Glossary
Geometry
The addition rule states that given events A and B, the probability of either A or B is given by \(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\).
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The addition rule states that given events A and B, the probability of either A or B is given by \(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\).
Arccosine is a relationship used to find an acute angle measure in a right triangle when two side lengths are known.
The arccosine of a number between 0 and 1 is the measure of an acute angle whose cosine is that number.
\(\arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right)=\theta\)
Arcsine is a relationship used to find an acute angle measure in a right triangle when two side lengths are known.
The arcsine of a number between 0 and 1 is the measure of an acute angle whose sine is that number.
\(\arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \theta\)
Arctangent is a relationship used to find an acute angle measure in a right triangle when two side lengths are known.
The arctangent of a positive number is the measure of an acute angle whose tangent is that number.
\(\arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right) = \theta\)
When a figure is circumscribed, it is completely surrounded by another figure, so their sides, edges, vertices, or curves touch.
Two figures are congruent if there is a rigid motion or a sequence of rigid motions (translations, rotations, and reflections) that takes one figure onto the other.
In this figure, Triangle A is congruent to Triangle D.
Corresponding parts are the matching parts of an original figure and its scaled copy that are in the same relative positions. The parts could be points, segments, angles, or distances. When two figures are congruent, all of their corresponding parts are congruent.
For example, in triangles \(ABC\) and \(DEF\):
A dilation is a transformation that can reduce or enlarge a figure.
A dilation with center \(P\) and positive scale factor \(k\) takes a point \(A\) along the ray \(PA\) to another point whose distance is \(k\) times farther away from \(P\) than \(A\) is.
Triangle \(A'B'C'\) is the result of applying a dilation with center \(P\) and scale factor 3 to triangle \(ABC\).
When a figure is inscribed, it is completely inside another figure so that their sides, edges, vertices, or curves touch.
A line of symmetry is a line that divides a figure into two parts that are mirror images of each other. When a figure is reflected across one of its lines of symmetry, it takes the figure onto itself.
These dashed lines show two lines of symmetry for a regular hexagon, and two lines of symmetry for the capital letter I.
An oblique solid is not exactly upright—it seems to lean over at an angle.
A prism is a three-dimensional figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a pentagonal prism.
rectangular prism
triangular prism
pentagonal prism
A pyramid is a three-dimensional figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a hexagonal pyramid.
square pyramid
pentagonal pyramid
A reflection is a rigid transformation that is defined by a line. It takes one point to another point that is the same distance from the given line, but on the other side. The segment from the original point to its image is perpendicular to the line of reflection.
In this figure, \(A\) is reflected across line \(m\), and \(A′\) is the image of \(A\) under the reflection.
Reflect \(A\) across line \(m\).
A right solid is exactly upright—it does not seem to lean over at an angle.
A rotation is a rigid transformation that is defined by a center, an angle, and a direction. It takes one point on a circle to another point, using a given center. The two radii—the one from the center to the original point and the one from the center to the image—make the angle of rotation.
In this figure, \(P′\) is the image of \(P\) after a counterclockwise rotation of \(t^\circ\) using the point \(O\) as the center.
In this figure, quadrilateral \(ABCD\) is rotated \(120^\circ\) counterclockwise using the point \(D\) as the center.
A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (This does not include rotations using angles, such as \(0^\circ\) and \(360^\circ\), that take every point on a figure back to its original position.)
This hexagon has rotation symmetry 60 degrees clockwise or counterclockwise around its center.
A translation is a rigid transformation that is defined by a directed line segment. It takes one point to another point so that:
In the figure shown here, \(A'\) is the image of \(A\) under the translation given by the directed line segment \(t\).
Trigonometric ratios relate the angles and sides of right triangles.
Three trigonometric ratios are sine, cosine, and tangent.
\(\sin(\theta)=\dfrac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\theta)=\dfrac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\theta)=\dfrac{\text{opposite}}{\text{adjacent}}\)