Mathematics Geometry Lesson Points, Lines, and Planes: by Owen Borville September 8, 2026
A Point is a specific location in space. A Line contains an infinite collection of points in one dimension. Two different points define a unique line, such as point A and point B.
Collinear points refer to three or more points inside the same line. Points outside this line are Non-collinear. Any two points are collinear. If three points are collinear, one of the points is between the other two points (A-B-C) where point B is between points A and C.
A Line segment is a portion of a line with two end points and finite length. A Segment is the set of all points between the two endpoints.
A Ray is a half-line, or a portion of a line that has an endpoint and extends without end in the other direction. The ray from point A to point B is the ray AB with an arrow pointing toward point B. Opposite rays are two rays that have a common end point but extend in opposite directions.
A Plane is a flat, boundless surface in space with two dimensions of length and width (no height). Any three non-collinear points define a unique plane. Any line and a point not on the line define a unique plane. Co-planar points are a set of points or lines that lie in the same plane. Non-co-planar points do not lie in the same plane.
Space is the infinite set of all points in three dimensions: length, width, and height.
Congruent line segments have the same measure. The Midpoint of a line segment is the point that divides the line into two congruent segments. The Bisector of a line segment is a line or line subset that intersects the segment at its midpoint.
_________________________________________
Geometry Angles:
Angles (∠) are two rays that share a common endpoint. The vertex is the common endpoint. Degrees are used to measure angles. Each degree is subdivided into minutes and seconds. Radians (rad) can also be used to measure angles. One complete revolution is (2ℼ) radians. A radian of an angle is the measure of the length of an arc of a circle cut off by that angle.
Converting between degrees and radians: one revolution is 360 degrees or 2𝜋 rad. One rad is 180 degrees/𝜋 = 1/2𝜋 revolution. One degree is 𝜋/180 rad = 1/360 revolution.
Congruent angles are two angles of the same measure ∠A = ∠B.
Types of Angles:
Zero Angle = 0 degrees. Two rays correspond.
Right Angle = 90 degrees. Two rays are perpendicular.
Straight Angle = 180 degrees. The two rays are opposite.
Acute Angle = greater than 0 degrees but less than 90 degrees.
Obtuse Angle = greater than 90 degrees but less than 180 degrees.
Oblique Angle = acute or obtuse
Adjacent Angles = are two angles that have a common vertex and common side.
Complementary Angles = pair of angles whose sum is 90 degrees.
Supplementary Angles = pair of angles whose sum is 180 degrees.
Linear Pair of Angles = two adjacent angles whose sum is a straight angle.
Vertical Angles are two intersecting lines or segments that form four angles (two pairs of opposing vertical angles). The vertical angles in each pair are congruent. Any two adjacent angles are supplementary. If one of the angles in the intersection of two lines is a right angle, then all four angles are right angles.
Two Parallel Lines Cut By A Transversal Line not parallel to either line creates eight angles or four sets of vertical angles with unique relationships. A transversal line is a line that crosses two or more other lines in the same plane at different points. When a transversal cuts across parallel lines (lines that never meet), it creates eight distinct angles with special mathematical relationships.
Corresponding angle pairs = on the same side of the transversal line and separated by one of the parallel lines. Corresponding angles are congruent.
Alternate interior angle pairs are two pairs between the parallel lines on opposite sides of the transversal that are congruent.
Alternate exterior angle pairs are two pairs outside the parallel lines on opposite sides of the transversal that are congruent.
Interior angle pairs are two pairs on the same side of the transversal where interior angles are supplementary.
An Angle Bisector or bisector of an angle is the ray that lies within the interior of an angle with its endpoint on the vertex of that angle if it creates two new angles of equal measure.
_____________________________________
Parallel Lines are two lines in the same plane that never intersect. For every point not on a line in a plane, there is one parallel line through that point. Parallel lines symbol: Line A || Line B or A || B.
Perpendicular Lines are two intersecting lines that form a right angle (four right angles). Perpendicular lines symbol: A ⟂ B. For every line and every point, there is a point through which a perpendicular line passes through the point whether the point is on the line or not.
Line Segments are labelled by their start point and end point (Ex. line segment AB is the line from point A to point B.
Congruent: Two line segments of the same length are congruent. Ex. AB = YZ.
Midpoint of a segment is the point halfway between the two endpoints.
Bisector of a segment is any line passing through the midpoint of the segment.
Perpendicular bisector is a unique line bisecting and perpendicular to a segment.
Equidistant points are points that are the same distance from two other points, set of points, lines, or reference point.
Distance from a point to a line is the length of the perpendicular from the point to the line.
Distance between two parallel lines is the distance of a perpendicular from any point on a parallel line to the other parallel line.
________________________________
Polygons: A Polygon is a flat, two-dimensional geometric shape made entirely of straight line segments (straight sides) connected at endpoints to form a closed path or closed loop.
Vertices (vertex) are the endpoints of a polygon and the segments are called sides. The number of vertices and sides are equal.
Angles of the polygon are the inside angles.
Adjacent sides share a vertex.
A Diagonal of a polygon is a segment whose endpoints are non-consecutive vertices.
Congruent Polygons have the same shape and size, including corresponding sides and angles.
Similar Polygons have the same shape, but one may be larger or smaller than the other. Corresponding angles are congruent and side lengths are proportional to each other. Similar polygons may have different side lengths but the side lengths correspond by a certain ratio (similitude).
The Perimeter of a polygon is the sum of the lengths of all of its sides.
Area of a polygon is a measure of how much space is enclosed inside the plane, or how many non-overlapping square units of area can fit inside the polygon.
A polygon must have at least three sides and three angles, which would also be a triangle.
3 sides/angles = triangle
4 sides/angles = quadrilateral
5 sides/angles = pentagon
6 sides/angles = hexagon
7 sides/angles = septagon
8 sides/angles = octagon
9 sides/angles = nonagon
10 sides/angles = decagon
12 sides/angles = dodecagon
Convex Polygons have angles all less than 180 degrees.
Concave polygons have at least one angle measuring more than 180 degrees.
Equilateral polygons have sides that are all congruent. Perimeter is the number of sides times the side length.
Equiangular polygons have angles that are all congruent and convex.
Regular polygons have sides and angles that are congruent and convex.
The center of a regular polygon is the point in the middle that is equidistant from all the vertices.
Apothems are segments connecting the center and the midpoint of a side in a regular polygon.
Perimeter (P) = (number of sides) x (side length).
Area (A) = 1/2P x (apothem length)
The Number of Diagonals of a polygon is n(n-3)/2.
Angle Sum of a Polygon is s = sum of n interior angles = (n-2)180 degrees.
Exterior Angles are the corresponding exterior angles of any interior angle of a polygon less than 180 degrees. In a convex polygon, the sum of all the exterior angles is 360 degrees.
_________________________________
Quadrilaterals are two dimensional closed polygons with four straight sides, four vertices, two diagonals, four angles, interior angle sum of 360 degrees. Types of quadrilaterals include convex and concave quadrilaterals, trapezoid, parallelogram, rectangle, square, and rhombus.
A Parallelogram is a quadrilateral where opposite sides are parallel and congruent. Opposite angles are congruent, diagonals bisect each other, interior angles sum to 360 degrees.
The Altitude of a parallelogram is a perpendicular line between a pair of opposite sides. Length of the altitude is the height of the parallelogram. The base of the parallelogram is the side to which the altitude is extended to.
Properties:
The Perimeter of a parallelogram = is two times the sum of the lengths of two unequal sides.
The Area of a parallelogram = is the base length times the height.
A Parallelogram has two diagonals, parallel sides, congruent sides, bisected diagonals, congruent angles, and supplementary angles.
_____________________________
A Rectangle is a special parallelogram where all four angles are right angles and equiangular.
A Rhombus is a special parallelogram where all four sides are congruent and equilateral.
A Square is a special parallelogram with four congruent sides and four congruent angles. A square is also a regular quadrilateral, rectangle, and rhombus.
_____________________________
Rectangle is a quadrilateral with four right angles and is a parallelogram. Opposite sides are parallel and congruent. Diagonals are congruent and bisect each other. All four angles are equal and 90 degrees.
The Perimeter of rectangle = P + 2((base) + (height))
Area of rectangle = (base) x (height)
_______________________________
Rhombus is a parallelogram with four congruent sides.
Properties: Opposite sides are parallel and all sides are congruent. Opposite angles are congruent. Diagonals are perpendicular to and bisect each other. Diagonals bisect angles.
Perimeter of rhombus = P = 4(side length)
Area of rhombus = ½ (long diagonal)x(short diagonal)
_____________________________________
A Square has four right angles and four equal sides. A square is a rectangle and rhombus.
Properties: Opposite sides are parallel and all sides are congruent. All angles are equal and measure 90 degrees.
Diagonals are perpendicular and congruent, bisecting each other and divide the square into four right angles with two 45 degree angles inside each angle. The length of a diagonal is the square root of 2 times side length.
Perimeter of square = P = 4(side length)
Area of square = (side length)^squared(2)
________________________________
A Trapezoid is a quadrilateral with exactly one pair of parallel sides.
The Base of a trapezoid is either of two parallel sides.
The Leg of a trapezoid is either of the other two (non-parallel) sides.
The Median of a trapezoid is a segment joining the midpoints of the two legs, parallel to the bases. Its length is the average of the lengths of the two bases.
The Altitude of a trapezoid is a segment perpendicular to the bases that joins a point on one base to the line that contains the other base.
Perimeter of trapezoid = P = sum of four side lengths.
Area of trapezoid = ½ (base1 + base2) x (altitude) = median x altitude
Isosceles trapezoid is a trapezoid with congruent legs (one pair of parallel sides and one pair of opposite congruent sides).
Properties: Legs are congruent. Median is parallel to the bases and its length is the average of the two base lengths. Diagonals are congruent. Lower base angles are congruent and the upper base angles are congruent. The two angles adjacent to a leg are supplementary.
Perimeter of an isosceles trapezoid = (base1) + (base2) + 2(leg)
Area of an isosceles trapezoid = ½ (base1 + base2) x (altitude)
________________________________
Triangles are polygons that have three sides, three vertices, and three angles. The sum of a triangle’s angles is 180 degrees.
Scalene triangle = three different side lengths and three different angle measures
Isosceles triangle = at least two congruent sides and two congruent angles
Equilateral triangle = three congruent sides and three congruent angles (each 60 degrees).
Acute triangle = three acute angles
Obtuse triangle = one obtuse angle
Right triangle = one right angle of 90 degrees
Equiangular triangle = has three congruent angles
_____________________________________
Congruence: Congruent triangles have the same shape and size (angles and sides are congruent).
Similar Triangles have the same shape, angles are congruent and sides are proportional.
Testing Triangle Congruence: Triangles have six attributes: three sides and three angles. Three attributes must be proven to establish congruence.
SSS = three sides of two triangles are congruent
SAS = two sides and the angle between them are congruent
ASA = two angles and the side between them are congruent
AAS = use ASA
_________________________________________
Triangle Base: one side of a triangle, usually a horizontal side.
Triangle Altitude is a perpendicular line segment from one vertex to the line that contains the opposite side. Height is the length of the altitude.
Triangle Median is the line drawn from the vertex to the midpoint of the opposite side.
Area of a triangle = A = ½ (base) x (height)
Area of a triangle (Heron’s area formula) when the lengths of the three sides is known.
Median area = triangle divided into two halves of equal area by a median line.
Length of a triangle side is always less than the sum of the two other sides lengths.
Two angles are equal in a triangle if and only if the sides opposite them are equal. If the angles are not equal, the longer side is opposite the larger angle. If the sides are unequal, the larger angle is opposite the longer side.
The exterior angle is equal to the sum of the measures of the other two interior angles. The exterior angle is always greater than the measure of either interior angle.
The angle bisector of a triangle is a line that bisects any angle of a triangle and ends on the opposite side of that angle.
The perpendicular bisector of a line segment is a line, line segment, or ray that is perpendicular to the line segment at its midpoint.
_____________________________________
Isosceles Triangle: the two equal sides of an isosceles triangle are called legs. The third side is the base.
Angles A and C = base angles of an isosceles triangle = Angle A = Angle C = ½(180 degrees - Angle B)
The altitude to the base bisects the angle opposite the base and hits the base at the midpoint. The altitude is also the median. The altitude also splits the isosceles triangle into two congruent right triangles.
Side lengths a (leg) and b (base) of an isosceles triangle, h (altitude):
= a = √b²/4 + h²
= b = 2√a² - h²
= h = √a² - b²/4
__________________________
Equilateral Triangle: An equilateral triangle is also isosceles. In addition, every equilateral triangle has the same shape. The altitude splits the triangle into two congruent 30-60-90 degree right triangles.
Height and area of an equilateral triangle:
Height
Area = A = (s²√3)/4
Altitude (h) = (s√3)/2
______________________________________
A Right Triangle has one right angle of 90 degrees. The side opposite the right angle is the longest side and is called the hypotenuse. The other two sides are called legs. The sum of the other two angles is 90 degrees.
Angle measure: Area = A = ½(leg1) x (leg2)
Pythagorean Theorem = a² + b² = c² (c = hypotenuse, a and b are legs)
Right Triangle Similarity: Altitude splits the right triangle into two smaller right triangles similar to each other.
Special Right Triangles:
30 - 60 - 90 degree triangles = half of an equilateral triangle: hypotenuse = 2(shorter leg).
Longer leg = √3(shorter leg); Area = A = √3/2(shorter side)²
45-45-90 degree triangles = half of a square; isosceles right triangle. Legs are equal. Hypotenuse = √2(leg); A = 1/2 (leg)²
Testing Triangle Similarity:
AA: If two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar.
SSS: If all three sides of one triangle are proportional to the sides of another triangle, the two triangles are similar.
SAS: If two sides of one triangle are proportional to two sides of another triangle, the angles between these sides are congruent, and the triangles are similar.
Cutting Similar Triangles: A line parallel to one of the sides of a triangle creates another similar triangle by cutting it off.
Similarity Measurement: If two triangles are proportional, the perimeter of both is related by the proportionality constant and the area of both is related by the square of the proportionality constant.
In similar triangles, the altitudes and medians are proportional to the sides.
__________________________________
Circles: A Circle is the set of all points equidistant from the center point.
The Radius is the distance from any point on the circle to the center.
Diameter is the line segment containing the radius and whose endpoints are on the circle (on each side).
Circumference is the perimeter of the circle, calculated as 2(π)r, where r is the radius and π is pi, a unique real number about 3.14159…
Area of the circle is A = π(r²), where r is the radius.
A Chord is a line segment joining two points on a circle.
A Secant is a line segment that intersects the circle at two points. A secant line contains a chord.
A Tangent is a line that intersects a circle at exactly one point.
The Point of Tangency is the point of intersection of a circle and a line tangent to it.
A Tangent line is always Perpendicular to the radius drawn to the point of tangency.
An Arc is a continuous section of a circle. Any two points on a circle define two arcs and any chord defines two arcs. Arcs denoted by the letter symbols of the endpoints, such as AB or CD with an arc symbol drawn above it. Arcs are measured in degrees by the central angle that they subtend. A circle is an arc of 360 degrees.
The Central Angle is the angle whose vertex is the center of the circle. Any central angle intercepts the circle at two points, defining the arc.
A Semicircle is a half-circle arc measuring 180 degrees.
A Major Arc is larger than a semicircle or greater than 180 degrees.
A Minor Arc is smaller than a semicircle or less than 180 degrees.
Arc Length of a circle with radius r and arc angle θ (theta) degrees is = θ /180 (π)(r)
A Sector is the region inside a circle bounded by a central angle and the arc it defines.
Area of a Sector is a sector whose arc measures θ degrees and has area θ/360 π(r²).
A Segment is a region inside a circle bounded by a chord and the arc it defines.
Area of a Segment = A = (area of sector) - (area of triangle)
Congruent circles have congruent radii.
Congruent arcs on the same circle or on different circles are equal in measure.
______________________________
Chord and Arc Theorems:
The triangle produced by a chord in a circle is isosceles.
Congruent chords are equidistant from the center of the circle.
Congruent chords in a circle produce congruent arcs.
Parallel chords in the same circle create congruent arcs between them.
____________________________________
Secant and Tangent Angles to Circles:
The measure of an inscribed angle is half the measure of the arc it intercepts.
A right angle inscribed in a circle produces a diameter as the hypotenuse.
The measure of an angle formed by a chord and a tangent is half the measure of the arc it intercepts.
The Angle measure of two secants intersecting a circle is half the sum of measures of the arcs intercepted by the vertical angles. If the vertex is outside the circle, the angle of a secant or tangent lines is half the difference of the measures of the arcs intercepted by the secant or tangent lines.
The product of the lengths of the segments of one chord is equal to the product of the lengths of the segment of the other chord.
When two secant segments share an endpoint outside the circle, the products of the lengths of each secant segment with its external part are equal.
When a secant segment and a tangent segment share an endpoint not on the circle, the length of the tangent segment squared is equal to the product of the secant segment and its external part.
Tangent segments that share an endpoint outside the circle are congruent.
A circle is inscribed in a polygon if all of the sides of the polygon are tangent to the circle.
A circle is circumscribed around a polygon if all of the vertices of the polygon are on the circle. The polygon is inscribed inside the circle.
_________________________________
Lines and Planes in Space:
Lines can exist in the same plane but also in different planes.
Two lines in the same plane can intersect or are parallel.
If two lines in space intersect, then they lie in the same plane.
If two lines in space are parallel, then they lie in the same plane.
Skew lines do not lie in the same plane because they don’t intersect or are not parallel.
Lines and planes in space always either intersect or are parallel. If they are parallel, they never intersect.
A line is perpendicular to a plane only if it is perpendicular to every line in the plane that goes through their point of intersection.
For every point in the plane, there is a unique line perpendicular to the plane that goes through that point.
A line and a point on the line determine a unique plane containing the point and perpendicular to the line.
_________________________________
Two planes either intersect or are parallel.
Two planes are parallel if and only if they never intersect. Any line perpendicular to one is perpendicular to both. The distance between the two planes is the distance between the points of intersection along a perpendicular line. Two non-parallel planes always intersect in a line.
Two intersecting planes are said to be perpendicular if one of the planes contains a line perpendicular to the other plane.
Dihedral angle is the angle formed by the union of two planes with a common edge, or the geometric angle formed between two intersecting planes or half-planes. Two perpendicular planes intersect to form a right dihedral angle.
____________________________________
Solids Shapes: A polyhedron is a solid region formed from the intersection of several planes (at least four). The planes intersect in polygonal faces whose sides are called edges and whose corners are the vertices of the polyhedron.
Surface area of the polyhedron is the total area of all of the faces of the polyhedron.
Volume of the polyhedron is how much space fits inside the solid figure in cubic units.
___________________________________________
Six-Faced Solids: Cubes, Rectangular solids, Parallelepipeds
Rectangular solids are polyhedrons with six rectangular faces. Adjacent faces intersect at right angles. Measurements of length, width, and height.
Volume rectangular solids = V = length x width x height
Surface Area rectangular solids = SA = 2 (lw + lh + lw)
____________________________________________
A Cube is a rectangular solid with six congruent square faces and has twelve congruent edges and eight vertices.
Volume of a cube = V = (side length)³
Surface area of a cube = SA = 6(side length)²
___________________________________________
A Parallelepiped is a polyhedron whose six faces are parallelograms lying in pairs of parallel planes. Rectangular solids are parallelepipeds whose adjacent faces lie in perpendicular planes. Volume of parallelepiped = V = lwh.
____________________________
A Prism is a polyhedron of which two faces (the bases) are congruent polygons lying in parallel planes. The other faces, lateral faces, are parallelograms that join corresponding sides on the congruent polygons. Lateral edges are where the sides join the lateral parallelograms to each other. A prism is defined by the shape of its bases.
Height of the prism is the perpendicular distance between the bases or planes containing them.
Volume of a prism is V = Base area x height
Lateral area of a prism is the area of the lateral faces of the prism.
A Right Prism is a prism whose lateral edges are perpendicular to the planes containing the bases of the prism.
A Right Triangular Prism is a type of right prism. Lateral area for a right prism = LA = perimeter of base x height
Parallelepipeds are prisms with a parallelogram base. Rectangular parallelepipeds are parallelepipeds with rectangular bases and lateral edges perpendicular to the bases.
Rectangular solids and cubes are right prisms with rectangular or square bases.
The Altitude of a prism is a line segment perpendicular to each of the bases with an endpoint on each base.
The total surface area of a solid object is the sum of the lateral area and the area of the bases.
__________________________________________________
A Cylinder is like a prism, except its bases are circular, and has congruent parallel curves.
The axis of a cylinder is the line connecting the centers of the two bases.
Volume of a cylinder = V = base area x height = (π r²) x height.
A Right cylinder is a cylinder whose axis is perpendicular to its bases or the planes containing them.
Lateral area = LA = perimeter of base x height = 2π(radius) x (height)
Surface area = SA = LA + 2(Base area) = 2π(radius)² + 2π(radius) x (height)
________________________________
A Pyramid is the set of all points along segments that join a polygonal base with a vertex not in the plane of the base. If the base is a polygon with n edges, the pyramid has n + 1 faces. One base and n triangular lateral faces.
The Height of the pyramid is the distance from the vertex to the base or the plane that contains the base.
Volume of the pyramid is V = ⅓ (base area) x (height)
A Regular pyramid is a pyramid with two properties (1) the base is a regular polygon (2) the line joining the vertex and the center of the base (altitude) is perpendicular to the base. All lateral faces are congruent isosceles triangles.
Slant height is the length of the altitude of one of the lateral faces, calculated by the Pythagorean Theorem (a² + b² = c²), with c being the slant height.
Lateral Area of a Pyramid = LA = ½ base perimeter x slant height
Surface Area of a Pyramid = SA = LA + Base Area
A Tetrahedron is a regular triangular pyramid with four triangular faces, four vertices, and six edges.
____________________________________________
A Cone is like a pyramid with a circular base.
Volume of Cone = ⅓ base area x height = ⅓ π(radius)² x (height)
Right Cone = is a cone whose axis is perpendicular to the plane containing the circular base.
Slant height is the length of the shortest segment from the vertex to a point on the perimeter of the circular base. The segment is perpendicular to the tangent to the circle at the point where the segment hits the circular base. Using the Pythagorian Theorem, (radius)² + (height)² = (slant height)²
Lateral Area = LA = ½ (base perimeter) x slant height = π(radius) x (slant height)
___________________________________
A Sphere is the three dimensional version of a circle and is the set of all points in space equidistant from a fixed point called the center and common distance from the center to the surface called the radius. A hemisphere is half of a sphere. A great circle of a sphere is a plane intersecting the sphere through the center of the sphere.
Volume of a sphere = V = 4/3π(radius)³
Surface Area of a sphere = SA = 4π(radius)²
__________________________________________
Transformation geometry is a branch of mathematics that studies how geometric figures change position, orientation, or size on a plane or coordinate system. Two sets of points on a coordinate system correspond to a geometric figure and its transformed version: an original set of points and a final set of points. These two sets of points correspond with each other. In this approach, the original shape is called the preimage, and the final shape after the change is called the image.
Main Types of Transformations:
Translation: Slides a figure a set distance in a specific direction without changing its size, shape, or orientation.
Rotation: Turns a figure around a fixed central point by a specific angle.
Reflection: Flips a figure over a specific line to create a mirror image.
Dilation: Resizes a figure by making it larger or smaller based on a scale factor while keeping its proportional shape. The center of the dilation is the same point afterward. Rays from the center point to the outer point are the same rays. A positive dilation keeps the new image on the same side of the center point as the original, while a negative dilation flips the new image to the opposite side of the center point.
The Projection of a point on a line is the closest point on that line to the given point, found by dropping a perpendicular line from the point down to the target line.
The Projection of a line segment on a line is the new segment formed by dropping perpendicular lines from the endpoints of the original segment onto the target line.
The Pythagorean Triple is = a squared plus b squared = c squared, a, b, and c can be side lengths of a right triangle.
Rigid vs. Non-Rigid Transformations: Rigid Transformations (Isometries): Translations, rotations, and reflections are rigid because they change only the position or direction of a shape while keeping its size and angles exactly the same. Non-Rigid Transformations: Dilations and shears change the actual size or proportions of the original shape.
Isometry is a transformation that preserves distance between any two points, regardless of whether orientation is preserved or not.
__________________________________________
The slope of a line is a measure of the rise over the run, or the steepness of a line (vertical change over horizontal change).
Concurrent lines are three or more straight lines that cross through the exact same single point.
The orthocenter of a triangle is the single point where all three of its altitudes intersect.
The incenter is the single point inside a triangle where all three of its internal angle bisectors intersect.
A centroid is the geometric center or average position of all points in a shape.
The circumcenter of a triangle is the point where the three perpendicular bisectors of its sides intersect.
______________________________________
The Ratio of two numbers x and y (y is not = 0) = (x/y) or (x:y)
Proportion in mathematics is an equation that equalizes two ratios. Ex. (y/z = a/b). Solved by cross-multiplication of outer (extreme) terms and inner terms (means): b x y = a x z.
A Line Segment is divided proportionally when it is divided into smaller parts according to a specific ratio.
_________________________________________________________
Loci or locus of points is a set of points that satisfy a particular condition. Examples of loci are circles, perpendicular bisectors, angle bisectors, parabolas, and ellipses. Applications of locus in the real world are with GPS, navigation, optics, acoustics, architecture, urban planning, robotics, astronomy of planetary orbits.
A Circle is the locus of all points that sit at a fixed distance (the radius) from a single center point.
A Perpendicular Bisector: is the locus of all points that stay an equal distance away from two fixed points.
An Angle Bisector is the locus of all points that stay an equal distance away from two intersecting lines.
A Parabola is the locus of all points that stay equidistant from a fixed point (the focus) and a fixed line (the directrix).
An Ellipse is the locus of all points where the sum of the distances to two fixed points (foci) is always the same.
The locus of points equidistant from two given parallel lines is a single straight line that is parallel to both original lines and located exactly midway between them.
The locus of points at a fixed distance from a line is a pair of lines each parallel to the first line and a fixed distance from the first line.
A Point is a specific location in space. A Line contains an infinite collection of points in one dimension. Two different points define a unique line, such as point A and point B.
Collinear points refer to three or more points inside the same line. Points outside this line are Non-collinear. Any two points are collinear. If three points are collinear, one of the points is between the other two points (A-B-C) where point B is between points A and C.
A Line segment is a portion of a line with two end points and finite length. A Segment is the set of all points between the two endpoints.
A Ray is a half-line, or a portion of a line that has an endpoint and extends without end in the other direction. The ray from point A to point B is the ray AB with an arrow pointing toward point B. Opposite rays are two rays that have a common end point but extend in opposite directions.
A Plane is a flat, boundless surface in space with two dimensions of length and width (no height). Any three non-collinear points define a unique plane. Any line and a point not on the line define a unique plane. Co-planar points are a set of points or lines that lie in the same plane. Non-co-planar points do not lie in the same plane.
Space is the infinite set of all points in three dimensions: length, width, and height.
Congruent line segments have the same measure. The Midpoint of a line segment is the point that divides the line into two congruent segments. The Bisector of a line segment is a line or line subset that intersects the segment at its midpoint.
_________________________________________
Geometry Angles:
Angles (∠) are two rays that share a common endpoint. The vertex is the common endpoint. Degrees are used to measure angles. Each degree is subdivided into minutes and seconds. Radians (rad) can also be used to measure angles. One complete revolution is (2ℼ) radians. A radian of an angle is the measure of the length of an arc of a circle cut off by that angle.
Converting between degrees and radians: one revolution is 360 degrees or 2𝜋 rad. One rad is 180 degrees/𝜋 = 1/2𝜋 revolution. One degree is 𝜋/180 rad = 1/360 revolution.
Congruent angles are two angles of the same measure ∠A = ∠B.
Types of Angles:
Zero Angle = 0 degrees. Two rays correspond.
Right Angle = 90 degrees. Two rays are perpendicular.
Straight Angle = 180 degrees. The two rays are opposite.
Acute Angle = greater than 0 degrees but less than 90 degrees.
Obtuse Angle = greater than 90 degrees but less than 180 degrees.
Oblique Angle = acute or obtuse
Adjacent Angles = are two angles that have a common vertex and common side.
Complementary Angles = pair of angles whose sum is 90 degrees.
Supplementary Angles = pair of angles whose sum is 180 degrees.
Linear Pair of Angles = two adjacent angles whose sum is a straight angle.
Vertical Angles are two intersecting lines or segments that form four angles (two pairs of opposing vertical angles). The vertical angles in each pair are congruent. Any two adjacent angles are supplementary. If one of the angles in the intersection of two lines is a right angle, then all four angles are right angles.
Two Parallel Lines Cut By A Transversal Line not parallel to either line creates eight angles or four sets of vertical angles with unique relationships. A transversal line is a line that crosses two or more other lines in the same plane at different points. When a transversal cuts across parallel lines (lines that never meet), it creates eight distinct angles with special mathematical relationships.
Corresponding angle pairs = on the same side of the transversal line and separated by one of the parallel lines. Corresponding angles are congruent.
Alternate interior angle pairs are two pairs between the parallel lines on opposite sides of the transversal that are congruent.
Alternate exterior angle pairs are two pairs outside the parallel lines on opposite sides of the transversal that are congruent.
Interior angle pairs are two pairs on the same side of the transversal where interior angles are supplementary.
An Angle Bisector or bisector of an angle is the ray that lies within the interior of an angle with its endpoint on the vertex of that angle if it creates two new angles of equal measure.
_____________________________________
Parallel Lines are two lines in the same plane that never intersect. For every point not on a line in a plane, there is one parallel line through that point. Parallel lines symbol: Line A || Line B or A || B.
Perpendicular Lines are two intersecting lines that form a right angle (four right angles). Perpendicular lines symbol: A ⟂ B. For every line and every point, there is a point through which a perpendicular line passes through the point whether the point is on the line or not.
Line Segments are labelled by their start point and end point (Ex. line segment AB is the line from point A to point B.
Congruent: Two line segments of the same length are congruent. Ex. AB = YZ.
Midpoint of a segment is the point halfway between the two endpoints.
Bisector of a segment is any line passing through the midpoint of the segment.
Perpendicular bisector is a unique line bisecting and perpendicular to a segment.
Equidistant points are points that are the same distance from two other points, set of points, lines, or reference point.
Distance from a point to a line is the length of the perpendicular from the point to the line.
Distance between two parallel lines is the distance of a perpendicular from any point on a parallel line to the other parallel line.
________________________________
Polygons: A Polygon is a flat, two-dimensional geometric shape made entirely of straight line segments (straight sides) connected at endpoints to form a closed path or closed loop.
Vertices (vertex) are the endpoints of a polygon and the segments are called sides. The number of vertices and sides are equal.
Angles of the polygon are the inside angles.
Adjacent sides share a vertex.
A Diagonal of a polygon is a segment whose endpoints are non-consecutive vertices.
Congruent Polygons have the same shape and size, including corresponding sides and angles.
Similar Polygons have the same shape, but one may be larger or smaller than the other. Corresponding angles are congruent and side lengths are proportional to each other. Similar polygons may have different side lengths but the side lengths correspond by a certain ratio (similitude).
The Perimeter of a polygon is the sum of the lengths of all of its sides.
Area of a polygon is a measure of how much space is enclosed inside the plane, or how many non-overlapping square units of area can fit inside the polygon.
A polygon must have at least three sides and three angles, which would also be a triangle.
3 sides/angles = triangle
4 sides/angles = quadrilateral
5 sides/angles = pentagon
6 sides/angles = hexagon
7 sides/angles = septagon
8 sides/angles = octagon
9 sides/angles = nonagon
10 sides/angles = decagon
12 sides/angles = dodecagon
Convex Polygons have angles all less than 180 degrees.
Concave polygons have at least one angle measuring more than 180 degrees.
Equilateral polygons have sides that are all congruent. Perimeter is the number of sides times the side length.
Equiangular polygons have angles that are all congruent and convex.
Regular polygons have sides and angles that are congruent and convex.
The center of a regular polygon is the point in the middle that is equidistant from all the vertices.
Apothems are segments connecting the center and the midpoint of a side in a regular polygon.
Perimeter (P) = (number of sides) x (side length).
Area (A) = 1/2P x (apothem length)
The Number of Diagonals of a polygon is n(n-3)/2.
Angle Sum of a Polygon is s = sum of n interior angles = (n-2)180 degrees.
Exterior Angles are the corresponding exterior angles of any interior angle of a polygon less than 180 degrees. In a convex polygon, the sum of all the exterior angles is 360 degrees.
_________________________________
Quadrilaterals are two dimensional closed polygons with four straight sides, four vertices, two diagonals, four angles, interior angle sum of 360 degrees. Types of quadrilaterals include convex and concave quadrilaterals, trapezoid, parallelogram, rectangle, square, and rhombus.
A Parallelogram is a quadrilateral where opposite sides are parallel and congruent. Opposite angles are congruent, diagonals bisect each other, interior angles sum to 360 degrees.
The Altitude of a parallelogram is a perpendicular line between a pair of opposite sides. Length of the altitude is the height of the parallelogram. The base of the parallelogram is the side to which the altitude is extended to.
Properties:
The Perimeter of a parallelogram = is two times the sum of the lengths of two unequal sides.
The Area of a parallelogram = is the base length times the height.
A Parallelogram has two diagonals, parallel sides, congruent sides, bisected diagonals, congruent angles, and supplementary angles.
_____________________________
A Rectangle is a special parallelogram where all four angles are right angles and equiangular.
A Rhombus is a special parallelogram where all four sides are congruent and equilateral.
A Square is a special parallelogram with four congruent sides and four congruent angles. A square is also a regular quadrilateral, rectangle, and rhombus.
_____________________________
Rectangle is a quadrilateral with four right angles and is a parallelogram. Opposite sides are parallel and congruent. Diagonals are congruent and bisect each other. All four angles are equal and 90 degrees.
The Perimeter of rectangle = P + 2((base) + (height))
Area of rectangle = (base) x (height)
_______________________________
Rhombus is a parallelogram with four congruent sides.
Properties: Opposite sides are parallel and all sides are congruent. Opposite angles are congruent. Diagonals are perpendicular to and bisect each other. Diagonals bisect angles.
Perimeter of rhombus = P = 4(side length)
Area of rhombus = ½ (long diagonal)x(short diagonal)
_____________________________________
A Square has four right angles and four equal sides. A square is a rectangle and rhombus.
Properties: Opposite sides are parallel and all sides are congruent. All angles are equal and measure 90 degrees.
Diagonals are perpendicular and congruent, bisecting each other and divide the square into four right angles with two 45 degree angles inside each angle. The length of a diagonal is the square root of 2 times side length.
Perimeter of square = P = 4(side length)
Area of square = (side length)^squared(2)
________________________________
A Trapezoid is a quadrilateral with exactly one pair of parallel sides.
The Base of a trapezoid is either of two parallel sides.
The Leg of a trapezoid is either of the other two (non-parallel) sides.
The Median of a trapezoid is a segment joining the midpoints of the two legs, parallel to the bases. Its length is the average of the lengths of the two bases.
The Altitude of a trapezoid is a segment perpendicular to the bases that joins a point on one base to the line that contains the other base.
Perimeter of trapezoid = P = sum of four side lengths.
Area of trapezoid = ½ (base1 + base2) x (altitude) = median x altitude
Isosceles trapezoid is a trapezoid with congruent legs (one pair of parallel sides and one pair of opposite congruent sides).
Properties: Legs are congruent. Median is parallel to the bases and its length is the average of the two base lengths. Diagonals are congruent. Lower base angles are congruent and the upper base angles are congruent. The two angles adjacent to a leg are supplementary.
Perimeter of an isosceles trapezoid = (base1) + (base2) + 2(leg)
Area of an isosceles trapezoid = ½ (base1 + base2) x (altitude)
________________________________
Triangles are polygons that have three sides, three vertices, and three angles. The sum of a triangle’s angles is 180 degrees.
Scalene triangle = three different side lengths and three different angle measures
Isosceles triangle = at least two congruent sides and two congruent angles
Equilateral triangle = three congruent sides and three congruent angles (each 60 degrees).
Acute triangle = three acute angles
Obtuse triangle = one obtuse angle
Right triangle = one right angle of 90 degrees
Equiangular triangle = has three congruent angles
_____________________________________
Congruence: Congruent triangles have the same shape and size (angles and sides are congruent).
Similar Triangles have the same shape, angles are congruent and sides are proportional.
Testing Triangle Congruence: Triangles have six attributes: three sides and three angles. Three attributes must be proven to establish congruence.
SSS = three sides of two triangles are congruent
SAS = two sides and the angle between them are congruent
ASA = two angles and the side between them are congruent
AAS = use ASA
_________________________________________
Triangle Base: one side of a triangle, usually a horizontal side.
Triangle Altitude is a perpendicular line segment from one vertex to the line that contains the opposite side. Height is the length of the altitude.
Triangle Median is the line drawn from the vertex to the midpoint of the opposite side.
Area of a triangle = A = ½ (base) x (height)
Area of a triangle (Heron’s area formula) when the lengths of the three sides is known.
Median area = triangle divided into two halves of equal area by a median line.
Length of a triangle side is always less than the sum of the two other sides lengths.
Two angles are equal in a triangle if and only if the sides opposite them are equal. If the angles are not equal, the longer side is opposite the larger angle. If the sides are unequal, the larger angle is opposite the longer side.
The exterior angle is equal to the sum of the measures of the other two interior angles. The exterior angle is always greater than the measure of either interior angle.
The angle bisector of a triangle is a line that bisects any angle of a triangle and ends on the opposite side of that angle.
The perpendicular bisector of a line segment is a line, line segment, or ray that is perpendicular to the line segment at its midpoint.
_____________________________________
Isosceles Triangle: the two equal sides of an isosceles triangle are called legs. The third side is the base.
Angles A and C = base angles of an isosceles triangle = Angle A = Angle C = ½(180 degrees - Angle B)
The altitude to the base bisects the angle opposite the base and hits the base at the midpoint. The altitude is also the median. The altitude also splits the isosceles triangle into two congruent right triangles.
Side lengths a (leg) and b (base) of an isosceles triangle, h (altitude):
= a = √b²/4 + h²
= b = 2√a² - h²
= h = √a² - b²/4
__________________________
Equilateral Triangle: An equilateral triangle is also isosceles. In addition, every equilateral triangle has the same shape. The altitude splits the triangle into two congruent 30-60-90 degree right triangles.
Height and area of an equilateral triangle:
Height
Area = A = (s²√3)/4
Altitude (h) = (s√3)/2
______________________________________
A Right Triangle has one right angle of 90 degrees. The side opposite the right angle is the longest side and is called the hypotenuse. The other two sides are called legs. The sum of the other two angles is 90 degrees.
Angle measure: Area = A = ½(leg1) x (leg2)
Pythagorean Theorem = a² + b² = c² (c = hypotenuse, a and b are legs)
Right Triangle Similarity: Altitude splits the right triangle into two smaller right triangles similar to each other.
Special Right Triangles:
30 - 60 - 90 degree triangles = half of an equilateral triangle: hypotenuse = 2(shorter leg).
Longer leg = √3(shorter leg); Area = A = √3/2(shorter side)²
45-45-90 degree triangles = half of a square; isosceles right triangle. Legs are equal. Hypotenuse = √2(leg); A = 1/2 (leg)²
Testing Triangle Similarity:
AA: If two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar.
SSS: If all three sides of one triangle are proportional to the sides of another triangle, the two triangles are similar.
SAS: If two sides of one triangle are proportional to two sides of another triangle, the angles between these sides are congruent, and the triangles are similar.
Cutting Similar Triangles: A line parallel to one of the sides of a triangle creates another similar triangle by cutting it off.
Similarity Measurement: If two triangles are proportional, the perimeter of both is related by the proportionality constant and the area of both is related by the square of the proportionality constant.
In similar triangles, the altitudes and medians are proportional to the sides.
__________________________________
Circles: A Circle is the set of all points equidistant from the center point.
The Radius is the distance from any point on the circle to the center.
Diameter is the line segment containing the radius and whose endpoints are on the circle (on each side).
Circumference is the perimeter of the circle, calculated as 2(π)r, where r is the radius and π is pi, a unique real number about 3.14159…
Area of the circle is A = π(r²), where r is the radius.
A Chord is a line segment joining two points on a circle.
A Secant is a line segment that intersects the circle at two points. A secant line contains a chord.
A Tangent is a line that intersects a circle at exactly one point.
The Point of Tangency is the point of intersection of a circle and a line tangent to it.
A Tangent line is always Perpendicular to the radius drawn to the point of tangency.
An Arc is a continuous section of a circle. Any two points on a circle define two arcs and any chord defines two arcs. Arcs denoted by the letter symbols of the endpoints, such as AB or CD with an arc symbol drawn above it. Arcs are measured in degrees by the central angle that they subtend. A circle is an arc of 360 degrees.
The Central Angle is the angle whose vertex is the center of the circle. Any central angle intercepts the circle at two points, defining the arc.
A Semicircle is a half-circle arc measuring 180 degrees.
A Major Arc is larger than a semicircle or greater than 180 degrees.
A Minor Arc is smaller than a semicircle or less than 180 degrees.
Arc Length of a circle with radius r and arc angle θ (theta) degrees is = θ /180 (π)(r)
A Sector is the region inside a circle bounded by a central angle and the arc it defines.
Area of a Sector is a sector whose arc measures θ degrees and has area θ/360 π(r²).
A Segment is a region inside a circle bounded by a chord and the arc it defines.
Area of a Segment = A = (area of sector) - (area of triangle)
Congruent circles have congruent radii.
Congruent arcs on the same circle or on different circles are equal in measure.
______________________________
Chord and Arc Theorems:
The triangle produced by a chord in a circle is isosceles.
Congruent chords are equidistant from the center of the circle.
Congruent chords in a circle produce congruent arcs.
Parallel chords in the same circle create congruent arcs between them.
____________________________________
Secant and Tangent Angles to Circles:
The measure of an inscribed angle is half the measure of the arc it intercepts.
A right angle inscribed in a circle produces a diameter as the hypotenuse.
The measure of an angle formed by a chord and a tangent is half the measure of the arc it intercepts.
The Angle measure of two secants intersecting a circle is half the sum of measures of the arcs intercepted by the vertical angles. If the vertex is outside the circle, the angle of a secant or tangent lines is half the difference of the measures of the arcs intercepted by the secant or tangent lines.
The product of the lengths of the segments of one chord is equal to the product of the lengths of the segment of the other chord.
When two secant segments share an endpoint outside the circle, the products of the lengths of each secant segment with its external part are equal.
When a secant segment and a tangent segment share an endpoint not on the circle, the length of the tangent segment squared is equal to the product of the secant segment and its external part.
Tangent segments that share an endpoint outside the circle are congruent.
A circle is inscribed in a polygon if all of the sides of the polygon are tangent to the circle.
A circle is circumscribed around a polygon if all of the vertices of the polygon are on the circle. The polygon is inscribed inside the circle.
_________________________________
Lines and Planes in Space:
Lines can exist in the same plane but also in different planes.
Two lines in the same plane can intersect or are parallel.
If two lines in space intersect, then they lie in the same plane.
If two lines in space are parallel, then they lie in the same plane.
Skew lines do not lie in the same plane because they don’t intersect or are not parallel.
Lines and planes in space always either intersect or are parallel. If they are parallel, they never intersect.
A line is perpendicular to a plane only if it is perpendicular to every line in the plane that goes through their point of intersection.
For every point in the plane, there is a unique line perpendicular to the plane that goes through that point.
A line and a point on the line determine a unique plane containing the point and perpendicular to the line.
_________________________________
Two planes either intersect or are parallel.
Two planes are parallel if and only if they never intersect. Any line perpendicular to one is perpendicular to both. The distance between the two planes is the distance between the points of intersection along a perpendicular line. Two non-parallel planes always intersect in a line.
Two intersecting planes are said to be perpendicular if one of the planes contains a line perpendicular to the other plane.
Dihedral angle is the angle formed by the union of two planes with a common edge, or the geometric angle formed between two intersecting planes or half-planes. Two perpendicular planes intersect to form a right dihedral angle.
____________________________________
Solids Shapes: A polyhedron is a solid region formed from the intersection of several planes (at least four). The planes intersect in polygonal faces whose sides are called edges and whose corners are the vertices of the polyhedron.
Surface area of the polyhedron is the total area of all of the faces of the polyhedron.
Volume of the polyhedron is how much space fits inside the solid figure in cubic units.
___________________________________________
Six-Faced Solids: Cubes, Rectangular solids, Parallelepipeds
Rectangular solids are polyhedrons with six rectangular faces. Adjacent faces intersect at right angles. Measurements of length, width, and height.
Volume rectangular solids = V = length x width x height
Surface Area rectangular solids = SA = 2 (lw + lh + lw)
____________________________________________
A Cube is a rectangular solid with six congruent square faces and has twelve congruent edges and eight vertices.
Volume of a cube = V = (side length)³
Surface area of a cube = SA = 6(side length)²
___________________________________________
A Parallelepiped is a polyhedron whose six faces are parallelograms lying in pairs of parallel planes. Rectangular solids are parallelepipeds whose adjacent faces lie in perpendicular planes. Volume of parallelepiped = V = lwh.
____________________________
A Prism is a polyhedron of which two faces (the bases) are congruent polygons lying in parallel planes. The other faces, lateral faces, are parallelograms that join corresponding sides on the congruent polygons. Lateral edges are where the sides join the lateral parallelograms to each other. A prism is defined by the shape of its bases.
Height of the prism is the perpendicular distance between the bases or planes containing them.
Volume of a prism is V = Base area x height
Lateral area of a prism is the area of the lateral faces of the prism.
A Right Prism is a prism whose lateral edges are perpendicular to the planes containing the bases of the prism.
A Right Triangular Prism is a type of right prism. Lateral area for a right prism = LA = perimeter of base x height
Parallelepipeds are prisms with a parallelogram base. Rectangular parallelepipeds are parallelepipeds with rectangular bases and lateral edges perpendicular to the bases.
Rectangular solids and cubes are right prisms with rectangular or square bases.
The Altitude of a prism is a line segment perpendicular to each of the bases with an endpoint on each base.
The total surface area of a solid object is the sum of the lateral area and the area of the bases.
__________________________________________________
A Cylinder is like a prism, except its bases are circular, and has congruent parallel curves.
The axis of a cylinder is the line connecting the centers of the two bases.
Volume of a cylinder = V = base area x height = (π r²) x height.
A Right cylinder is a cylinder whose axis is perpendicular to its bases or the planes containing them.
Lateral area = LA = perimeter of base x height = 2π(radius) x (height)
Surface area = SA = LA + 2(Base area) = 2π(radius)² + 2π(radius) x (height)
________________________________
A Pyramid is the set of all points along segments that join a polygonal base with a vertex not in the plane of the base. If the base is a polygon with n edges, the pyramid has n + 1 faces. One base and n triangular lateral faces.
The Height of the pyramid is the distance from the vertex to the base or the plane that contains the base.
Volume of the pyramid is V = ⅓ (base area) x (height)
A Regular pyramid is a pyramid with two properties (1) the base is a regular polygon (2) the line joining the vertex and the center of the base (altitude) is perpendicular to the base. All lateral faces are congruent isosceles triangles.
Slant height is the length of the altitude of one of the lateral faces, calculated by the Pythagorean Theorem (a² + b² = c²), with c being the slant height.
Lateral Area of a Pyramid = LA = ½ base perimeter x slant height
Surface Area of a Pyramid = SA = LA + Base Area
A Tetrahedron is a regular triangular pyramid with four triangular faces, four vertices, and six edges.
____________________________________________
A Cone is like a pyramid with a circular base.
Volume of Cone = ⅓ base area x height = ⅓ π(radius)² x (height)
Right Cone = is a cone whose axis is perpendicular to the plane containing the circular base.
Slant height is the length of the shortest segment from the vertex to a point on the perimeter of the circular base. The segment is perpendicular to the tangent to the circle at the point where the segment hits the circular base. Using the Pythagorian Theorem, (radius)² + (height)² = (slant height)²
Lateral Area = LA = ½ (base perimeter) x slant height = π(radius) x (slant height)
___________________________________
A Sphere is the three dimensional version of a circle and is the set of all points in space equidistant from a fixed point called the center and common distance from the center to the surface called the radius. A hemisphere is half of a sphere. A great circle of a sphere is a plane intersecting the sphere through the center of the sphere.
Volume of a sphere = V = 4/3π(radius)³
Surface Area of a sphere = SA = 4π(radius)²
__________________________________________
Transformation geometry is a branch of mathematics that studies how geometric figures change position, orientation, or size on a plane or coordinate system. Two sets of points on a coordinate system correspond to a geometric figure and its transformed version: an original set of points and a final set of points. These two sets of points correspond with each other. In this approach, the original shape is called the preimage, and the final shape after the change is called the image.
Main Types of Transformations:
Translation: Slides a figure a set distance in a specific direction without changing its size, shape, or orientation.
Rotation: Turns a figure around a fixed central point by a specific angle.
Reflection: Flips a figure over a specific line to create a mirror image.
Dilation: Resizes a figure by making it larger or smaller based on a scale factor while keeping its proportional shape. The center of the dilation is the same point afterward. Rays from the center point to the outer point are the same rays. A positive dilation keeps the new image on the same side of the center point as the original, while a negative dilation flips the new image to the opposite side of the center point.
The Projection of a point on a line is the closest point on that line to the given point, found by dropping a perpendicular line from the point down to the target line.
The Projection of a line segment on a line is the new segment formed by dropping perpendicular lines from the endpoints of the original segment onto the target line.
The Pythagorean Triple is = a squared plus b squared = c squared, a, b, and c can be side lengths of a right triangle.
Rigid vs. Non-Rigid Transformations: Rigid Transformations (Isometries): Translations, rotations, and reflections are rigid because they change only the position or direction of a shape while keeping its size and angles exactly the same. Non-Rigid Transformations: Dilations and shears change the actual size or proportions of the original shape.
Isometry is a transformation that preserves distance between any two points, regardless of whether orientation is preserved or not.
__________________________________________
The slope of a line is a measure of the rise over the run, or the steepness of a line (vertical change over horizontal change).
Concurrent lines are three or more straight lines that cross through the exact same single point.
The orthocenter of a triangle is the single point where all three of its altitudes intersect.
The incenter is the single point inside a triangle where all three of its internal angle bisectors intersect.
A centroid is the geometric center or average position of all points in a shape.
The circumcenter of a triangle is the point where the three perpendicular bisectors of its sides intersect.
______________________________________
The Ratio of two numbers x and y (y is not = 0) = (x/y) or (x:y)
Proportion in mathematics is an equation that equalizes two ratios. Ex. (y/z = a/b). Solved by cross-multiplication of outer (extreme) terms and inner terms (means): b x y = a x z.
A Line Segment is divided proportionally when it is divided into smaller parts according to a specific ratio.
_________________________________________________________
Loci or locus of points is a set of points that satisfy a particular condition. Examples of loci are circles, perpendicular bisectors, angle bisectors, parabolas, and ellipses. Applications of locus in the real world are with GPS, navigation, optics, acoustics, architecture, urban planning, robotics, astronomy of planetary orbits.
A Circle is the locus of all points that sit at a fixed distance (the radius) from a single center point.
A Perpendicular Bisector: is the locus of all points that stay an equal distance away from two fixed points.
An Angle Bisector is the locus of all points that stay an equal distance away from two intersecting lines.
A Parabola is the locus of all points that stay equidistant from a fixed point (the focus) and a fixed line (the directrix).
An Ellipse is the locus of all points where the sum of the distances to two fixed points (foci) is always the same.
The locus of points equidistant from two given parallel lines is a single straight line that is parallel to both original lines and located exactly midway between them.
The locus of points at a fixed distance from a line is a pair of lines each parallel to the first line and a fixed distance from the first line.