Glossary
Geometry
The addition rule states that given events A and B, the probability of either A or B is given by \(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\).
An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.
In this diagram, the dashed line segments show the altitude of each triangle.
An angle bisector is a line through the vertex of an angle that divides it into two congruent angles.
In this diagram, the dashed line is the angle bisector.
An apex is the single point on a cone or pyramid that is farthest from the base. For a pyramid, the apex is where all the triangular faces meet.
An arc is a part of a circle’s circumference between two points on the circle.
In this diagram, the blue highlighting shows one arc of the circle.
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.
\(\arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right) = \theta\)
An auxiliary line is an extra line drawn on a figure to show hidden structure.
- The auxiliary line drawn on this isosceles triangle shows a line of symmetry.
- The auxiliary lines drawn on this parallelogram suggest a way of rearranging it into a rectangle.
An axis of rotation is a line that a two-dimensional figure is rotated around to produce a three-dimensional figure, called a solid of rotation.
In this diagram, the dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.
Cavalieri’s Principle states that if two solids of equal height are cut into cross-sections by parallel planes, and the corresponding cross-sections on each plane always have equal areas, then the two solids have the same volume.
A central angle is an angle formed by two radii of a circle that each have an endpoint at the center of the circle.
In this diagram, the angle arc symbol shows one central angle of the circle.
A chance experiment is something that can be done over and over again, and what will happen each time is not known.
For example, each time the spinner is spun, it could land on red, yellow, blue, or green.
A circle of radius \(r\) with center \(O\) is the set of all points that are a distance \(r\) units from \(O\).
To draw a circle of radius 3 units and center \(O\), use a compass to draw all the points at a distance 3 units from \(O\).
The circumcenter of a triangle is the intersection point of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle.
In this diagram, the circumcenter is point \(D\).
When a figure is circumscribed, it is completely surrounded by another figure, so their sides, edges, vertices, or curves touch.
- A polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.
- A circle is circumscribed by a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.
Two angles are complementary to each other if their measures add up to \(90^\circ\). The two acute angles in a right triangle are complementary to each other.
Conditional probability is the likelihood that one event occurs given that another event occurs.
A cone is a three-dimensional figure with a circular base and a point not in the plane of the base called the apex. Each point on the base is connected to the apex by a line segment.
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.
- A translation takes Triangle A onto Triangle B.
- A rotation takes Triangle B onto Triangle C.
- A reflection takes Triangle C onto Triangle D.
The converse of an if-then statement is the statement that switches the hypothesis and the conclusion.
Example:
- Statement: “If it is raining, then the game is canceled.”
- Converse: “If the game is canceled, then it is raining.”
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 \(DEF\):
- Point \(B\) corresponds to point \(E\).
- Segment \(AB\) corresponds to segment \(DE\).
- Angle \(BCA\) corresponds to angle \(EFD\).
The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse.
In this diagram, \(\cos(x)=\frac{b}{c}\).
A cross-section is the two-dimensional figure formed by intersecting a solid with a plane.
These diagrams show that the shape of the cross-section depends on how the plane intersects the solid.
The cube root of a number \(x\), written \(\sqrt[3]{x}\), is the number \(y\) whose cube is \(x\).
That is, \(y^3 = x\). So, \(\sqrt[3]{x}=y\).
Example: \(2^3 = 8\). So, \(\sqrt[3]{8}=2\).
If a circle can be drawn around a quadrilateral so that it passes through every vertex, the quadrilateral is called a cyclic quadrilateral.
That is, a cyclic quadrilateral can be circumscribed by a circle.
A cylinder is a three-dimensional figure with two parallel, congruent, circular bases, formed by translating one base to the other. Each pair of corresponding points on the bases is connected by a line segment.
Density is a measure of how tightly the amount of matter in a substance is packed into the space it takes up. That is, density is the mass of a substance per unit volume.
\(\text{density}=\frac{\text{mass}}{\text{volume}}\)
Events are called dependent when they are from the same experiment and where the outcome of one event affects the probability of another.
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 \(ABC\).
A directed line segment is a line segment that has distance (length) and direction.
The arrow on this directed line segment shows that it starts at point \(A\) and ends at point \(B\).
A directrix is the line that, together with a point called the focus, defines a parabola.
This diagram shows a parabola is the set of points equidistant from the focus and directrix.
An event is a set of 1 or more outcomes in a chance experiment.
For example, if a number cube is rolled, there are 6 possible outcomes.
Some events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.”
Any flat surface on a three-dimensional figure is a face.
- A cube has 6 faces that are all squares.
- A cylinder has 2 faces that are both circles.
A focus is the point that, together with a line called the directrix, defines a parabola.
This diagram shows a parabola is the set of points equidistant from the focus and directrix.
An image is the result of a transformation. Every part of the original figure moves in the same way to match up with a part of the image.
This diagram shows a transformation that takes \(A\) to \(A’\).
\(A\) is the original, and \(A’\) is the image.
The incenter of a triangle is the intersection point of all three of the triangle’s angle bisectors. It is the center of the triangle’s inscribed circle.
In this diagram the incenter is point \(D\).
Events are called independent when they are from the same experiment and the outcome of one event does not affect the probability of another.
When a figure is inscribed, it is completely inside another figure so that their sides, edges, vertices, or curves touch.
- A polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle.
- A circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.
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.
A median is a line drawn from a vertex of a triangle to the midpoint of the opposite side.
Each dashed line segment in this image is a median.
An oblique solid is not exactly upright—it seems to lean over at an angle.
- Prisms and cylinders are said to be oblique if the directed line segment that defines the translation between the bases is not perpendicular to the bases.
- A cone is said to be oblique if a line drawn from its apex at a right angle to the plane of its base does not intersect the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.
Two numbers are opposites if they are the same distance from 0 on the number line, but on opposite sides. One is negative, and the other is positive.
An outcome of a chance experiment is one of the things that can happen.
For example, the possible outcomes of tossing a coin are heads and tails.
A parabola is the set of points that are equidistant from a given point, called the focus, and a given line, called the directrix.
Two lines that never intersect are called parallel. Line segments can also be parallel if they extend into parallel lines.
This figure shows two parallel line segments.
A parallelogram is a type of quadrilateral that has 2 pairs of parallel sides.
Here are 2 examples of parallelograms.
The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to the segment.
In this diagram, the dashed line is the perpendicular bisector of segment \(AB\).
Point-slope form is one way to write the equation of a line using its slope and the coordinates of one point on the line.
For a line with slope \(m\) through the point \((h,k)\), point-slope form is usually written as \(y-k=m(x-h)\). It can also be written as \(y = k+m(x-h)\).
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
The probability of a chance event is a number from 0 to 1 that expresses the likelihood of the event occurring, with 0 meaning it will never occur and 1 meaning it will always occur.
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 radian is a unit of measurement for angles, based on the radius of a circle.
The radian measure of any central angle is this ratio: \(\frac{\text{length of intercepted arc}}{\text{radius}}\)
Two numbers that multiply to equal 1 are reciprocals.
If \(p\) is a rational number that is not 0, then the reciprocal of \(p\) is the number \(\frac{1}{p}\).
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 figure has reflection symmetry if there is a reflection that takes the figure onto itself.
In this diagram, the letter X has reflection symmetry over each of the dashed lines.
A regular polygon is a polygon where all of the sides are congruent and all of the angles are congruent.
A right solid is exactly upright—it does not seem to lean over at an angle.
- Prisms or cylinders are said to be right if the directed line segment that defines the translation between the bases is perpendicular to the bases.
- A cone is said to be right if a line drawn from its apex at a right angle to the plane of its base passes through the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.
A rigid transformation is a move that does not change any measurements of a figure.
Translations, rotations, and reflections are rigid motions. So is any sequence of any of these.
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.
The sample space is the list of every possible outcome for a chance experiment.
For example, the sample space for tossing two coins is:
| heads-heads | tails-heads |
| heads-tails | tails-tails |
To create a scaled copy of a figure, all the side lengths in the original figure are multiplied by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).
A sector is the region inside a circle between two radii.
In this diagram, the shaded region shows one sector of the circle.
One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.
Triangle \(A'B'C'\) is similar to triangle \(ABC\) because a rotation with center \(B\) followed by a dilation with center \(P\) takes \(ABC\) to \(A'B'C'\).
The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse.
In this diagram, \(\sin(x) = \frac{a}{c}.\)
A solid of rotation is a three-dimensional figure formed by rotating a two-dimensional figure around a line called the axis of rotation.
In this diagram, the axis of rotation is the dashed line. The green triangle is rotated around the axis of rotation to form a solid of rotation.
A sphere is the set of all points in a three-dimensional space that are the same distance from a center point. All the cross-sections of a sphere are circles.
A figure has symmetry if there is a rigid transformation that takes it onto itself. (This does not include transformations that take every point back to its original position.)
The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg.
In this diagram, \(\tan(x) = \frac{a}{b}.\)
A line is tangent to a circle if the line intersects the circle at exactly one point and is perpendicular to the radius.
A translation is a rigid transformation that is defined by a directed line segment. It takes one point to another point so that:
- The directed line segment from the original point to its image is parallel to the given directed line segment.
- Both directed line segments have the same length and direction.
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.
\(\sin(\theta)=\dfrac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\theta)=\dfrac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan(\theta)=\dfrac{\text{opposite}}{\text{adjacent}}\)