Glossary

  • addition rule

    The addition rule states that given events A and B, the probability of either A or B is given by \(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\).

  • altitude

    An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.

  • angle bisector

    A line through the vertex of an angle that divides it into two equal angles.

  • apex

    The single point on a cone or pyramid that is the furthest from the base. For a pyramid, the apex is where all the triangular faces meet.

  • arc

    The part of a circle lying between two points on the circle.

  • arccosine

    The arccosine of a number between 0 and 1 is the acute angle whose cosine is that number.

  • arcsine

    The arcsine of a number between 0 and 1 is the acute angle whose sine is that number.

  • arctangent

    The arctangent of a positive number is the acute angle whose tangent is that number.

  • assertion

    A statement that you think is true but have not yet proved.

  • auxiliary line

    An extra line drawn in a figure to reveal hidden structure. 

    For example, the line shown in the isosceles triangle is a line of symmetry, and the lines shown in the parallelogram suggest a way of rearranging it into a rectangle.

  • axis of rotation

    A line about which a two-dimensional figure is rotated to produce a three-dimensional figure, called a solid of rotation. The dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.

    Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.
  • Cavalieri’s Principle

    If two solids 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.

  • central angle

    An angle formed by two rays whose endpoints are the center of a circle.

  • chance experiment

    A chance experiment is something you can do over and over again, and you don’t know what will happen each time.

    For example, each time you spin the spinner, it could land on red, yellow, blue, or green.

    A spinner
  • chord

    A chord of a circle is a line segment both of whose endpoints are on the circle.

  • circle

    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 and center \(O\), use a compass to draw all the points at a distance 3 from \(O\).

  • circumcenter

    The circumcenter of a triangle is the intersection of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle.

  • circumscribed

    We say a polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.

  • complementary

    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

    The probability that one event occurs under the condition that another event occurs.

  • cone

    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.

  • congruent

    One figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second.

  • conjecture

    A reasonable guess that you are trying to either prove or disprove.

  • converse

    The converse of an if-then statement is the statement that interchanges the hypothesis and the conclusion. For example, the converse of "if it's Tuesday, then this must be Belgium" is "if this is Belgium, then it must be Tuesday."

  • corresponding

    For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. We also talk about corresponding parts when we are trying to prove two figures are congruent and set up a correspondence between the parts to see if the parts are congruent.

    In the figure, segment \(AB\) corresponds to segment \(DE\), and angle \(BCA\) corresponds to angle \(EFD\).

  • cosine

    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 the diagram, \(\cos(x)=\frac{b}{c}\).

  • cross section

    The figure formed by intersecting a solid with a plane.

     

     

  • cube root

    The cube root of a number \(x\), written \(\sqrt[3]{x}\), is the number \(y\) whose cube is \(x\). That is, \(y^3 = x\).

  • cyclic quadrilateral

    A quadrilateral whose vertices all lie on the same circle.

  • cylinder

    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

    The mass of a substance per unit volume.

  • dependent events

    Dependent events are two events from the same experiment for which the probability of one event depends on whether the other event happens.

  • dilation

    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 \(ABC\).

  • directed line segment

    A line segment with an arrow at one end specifying a direction.

  • directrix

    The line that, together with a point called the focus, defines a parabola, which is the set of points equidistant from the focus and directrix.

  • event

    An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes.

    An image showing the six different sides of a number cube

    Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.”

  • face

    Any flat surface on a three-dimensional figure is a face.

    A cube has 6 faces.

  • focus

    The point that, together with a line called the directrix, defines a parabola, which is the set of points equidistant from the focus and directrix.

  • image

    If a transformation takes \(A\) to \(A'\), then \(A\) is the original and \(A'\) is the image.

  • incenter

    The incenter of a triangle is the intersection of all three of the triangle’s angle bisectors. It is the center of the triangle’s inscribed circle.

  • independent events

    Independent events are two events from the same experiment for which the probability of one event is not affected by whether the other event occurs or not.

  • inscribed

    We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

     

  • inscribed angle

    An angle formed by two chords in a circle that share an endpoint.

  • line of symmetry

    A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself.

    The figure shows two lines of symmetry for a regular hexagon, and two lines of symmetry for the letter I.

  • line segment

    A set of points on a line with two endpoints.

  • median (geometry)

    A line from a vertex of a triangle to the midpoint of the opposite side. Each dashed line in the image is a median.

  • oblique (solid)

    Prisms and cylinders are said to be oblique if when one base is translated to coincide with the other, the directed line segment that defines the translation 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.

  • opposite

    Two numbers are opposites of each other if they are the same distance from 0 on the number line, but on opposite sides.

    The opposite of 3 is -3 and the opposite of -5 is 5.

  • outcome

    An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails.

  • parabola

    A parabola is the set of points that are equidistant from a given point, called the focus, and a given line, called the directrix.

  • parallel

    Two lines that don't intersect are called parallel. We can also call segments parallel if they extend into parallel lines.

  • parallelogram

    A quadrilateral in which pairs of opposite sides are parallel.

  • perpendicular bisector

    The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it. 

  • point-slope form

    The form of an equation 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 line with point h comma k on an x y axis.
  • prism

    A prism is a solid 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

  • probability

    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.

  • pyramid

    A pyramid is a solid 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

  • radian

    The radian measure of an angle whose vertex is at the center of a circle is the ratio between the length of the arc defined by the angle and the radius of the circle.

  • reciprocal

    If \(p\) is a rational number that is not zero, then the reciprocal of \(p\) is the number \(\frac{1}{p}\).

  • rectangle

    A quadrilateral with four right angles.

  • reflection

    A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line. 

    In the figure, \(A'\) is the image of \(A\) under the reflection across the line \(m\).

    Reflect $A$ across line $m$.

  • reflection symmetry

    A figure has reflection symmetry if there is a reflection that takes the figure to itself.

  • regular polygon

    A polygon where all of the sides are congruent and all the angles are congruent.

  • rhombus

    A quadrilateral with four congruent sides.

  • right (solid)

    Prisms or cylinders are said to be right if when one base is translated to coincide with the other, the directed line segment that defines the translation 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.

  • rigid transformation

    A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.

  • rotation

    A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle.

    \(P'\) is the image of \(P\) after a counterclockwise rotation of  \(t^\circ\) using the point \(O\) as the center.

    ​​​​​Quadrilateral \(ABCD\) is rotated 120 degrees counterclockwise using the point \(D\) as the center.

  • rotation symmetry

    A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (We don't count rotations using angles such as \(0^\circ\) and \(360^\circ\) that leave every point on the figure where it is.)

  • sample space

    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
  • scale factor

    The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.

  • sector

    The region inside a circle lying between two radii of the circle.

  • similar

    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 \(A'B'C'\).

    Triangle ABC, rotated and then dilated.
  • sine

    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 the diagram, \(\sin(x) = \frac{a}{c}.\)

  • solid of rotation

    A three-dimensional figure formed by rotating a two-dimensional figure using a line called the axis of rotation.

    The axis of rotation is the dashed line. The green triangle is rotated about the axis of rotation line to form a solid of rotation.

    Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.
  • sphere

    A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.

  • symmetry

    A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is).

  • tangent

    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 the diagram, \(\tan(x) = \frac{a}{b}.\)

  • tangent (line)

    A line is tangent to a circle if the line intersects the circle at exactly one point.

  • tessellation

    An arrangement of figures that covers the entire plane without gaps or overlaps.

  • theorem

    A statement that has been proved mathematically. 

  • translation

    A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction.

    In the figure, \(A'\) is the image of \(A\) under the translation given by the directed line segment \(t\).

  • trigonometric ratio

    Sine, cosine, and tangent are called trigonometric ratios.