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62 Cards in this Set

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  • Back
ratio
quotient of two numbers
proportion
an equation stating that two or more ratios are equal
extremes
the first and fourth terms in a proportion
means
the second and third terms in a proportion
Means-Extremes Products Theorem
In a proportion, the product of the the means is equal to the product of the extremes.
Means-Extremes Ratio Theorem
If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion.
geometric mean
equal means in a proportion
Found by finding the square root of the product of the extremes.
arithmetic mean
add the two extremes together and divide the sum by two
dilation
enlargement by a fixed proportion
reduction
reduction of size by a fixed proportion
def. similar polygons
1. The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.
2. The ratio of any pair of corresponding sides to the ratio of another pair of corresponding sides is equal.
3. The corresponding angles of similar polygons are equal.
AAA Postulate
If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar.
AA Theorem
If there exists a correspondence between the vertices of two triangles such that two angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar.
SSS~
If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of corresponding sides are equal, then the triangles are similar.
SAS~
If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar.
dihedral angle
If there are 2 given noncoplanar half planes with a common edge at AB, the union of these two half planes and their common edge is called a dihedral angle.
Altitude on Hypotenuse Theorem
If an altitude is drawn to the hypotenuse of a right triangle, then:
a. The two triangles formed are similar to the given right triangle and to each other.
b. The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse.
c. Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg.
Pythagorean Theorem
The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs.
Converse of the Pythagorean Theorem
If the square of the measure of one side of a triangle equals the sum of the squares of the measures of the other two sides, then the angle opposite the longest side is a right angle.
Distance Formula
Line PQ = √([x2 – x1]2 + [y2 – y1]2)
Pythagorean Triple
Any three whole numbers that satisfy the equation (a)squared + (b)squared = (c)squared
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
30-60-90 Triangle Theorem
In a triangle whose angles have the measures 30, 60, and 90, the lengths of the sides opposite these angles can be represented by x, x√3, and 2x.
45-45-90 Triangle Theorem
In a triangle whose angles have the measures 45, 45, and 90, the lengths of the sides opposite these angles can be represented by x, x, and x√.
circle
a set of all points in a plane that are a given distance from a given point in the plane
center of a circle
the given point of which a set of all points in a plane are a given distance away from
radius
the given distance that a set of all points in a plane are from a given point
concentric circles
two or more coplanar circles with the same center
congruent circles
two circles are congruent if they have congruent radii
interior of a circle
a point is in the interior of a circle if its distance from the center is less than the radius
exterior of a circle
a point is in the exterior of a circle if its distance from the center is greater than the radius
on a circle
a point is on a circle if its distance from the center is equal to the radius
chord
a segment joining any two points on the circle
diameter
a chord that passes through the center of the circle
distance from the center of a circle to a chord
the measure of the perpendicular segment from the center to the chord
radius-chord perpendicular bisector theorem
if a radius is perpendicular to a chord, then it bisects the chord
converse of radius-chord perpendicular bisector theorem
if a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord
chord perpendicular bisector theorem
the perpendicular bisector of a chord passese through the center of the circle
chord equidistance theorem
if two chords of a circle are equidistant from the center, then they are congruent
converse of chord equidistance theorem
if two chords of a circle are congruent, then they are equidistant from the center of the circle
central angles-intercepted arcs theorem
if two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent
central angles-corresponding chords theorem
if two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent
arcs-corresponding chords theorem
if two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent
secant
a line that intersects a circle at exactly two points
tangent
a line that intersects a circle at exactly one point
point of tangency
the point in which a tangent intersects a circle
tangent-radius postulate
a tangent line is perpendicular to the radius drawn to the point of contact
converse of tangent-radius postulate
if a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle
tangent segment
the part of a tangent line between the point of contact and a point outside the circle
secant segment
the part of a secant line that joins a point outside the circle to the farther intersection point of the secant and the circle
external part of a secant segment
the part of a secant line that joins the outside point to the nearer intersection point
two-tangent theorem
if two tangent segments are drawn to a circle from an external point, then those segments are congruent
tangent circles
circles that intersect each other at exactly one point
externally tangent circles
two circles are externally tangent if each of the tangent circles lies outside the other
internally tangent circles
two circles are internally tangent if one of the tangent circles lies inside the other
common tangent
a line tangnet to two circles
common internal tangent
a common tangent that lies between the two circles
common external tangent
a common tangent that doesn't lie between the two circles
inscribed angle
an angle whose vertex is on a circle and whose sides are determined by two chords
tangent-chord angle
an angle whose vertex is on a circle and whose sides are determined by a tangent and a chord that intersect at the tangent's point of contact
inscribed angle theorem
the measure of an inscribed angle or a tangent-chord angle is one-half the measure of its intercepted arc
chord-chord angle
an angle formed by two chords that intersect inside a circle but not at the center
chord-chord angle theorem
the measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle