7.8: Chapter 7 Review
Difficulty Level: At Grade
Created by: CK12
Keywords and Theorems
 Ratio
 A way to compare two numbers.
 Proportion
 When two ratios are set equal to each other.
 Means
 Mean (also called the arithmetic mean): The numerical balancing point of the data set. Calculated by adding all the data values and dividing the sum by the total number of data points.
 Extremes
 the product of the means must equal the product of the extremes.
 CrossMultiplication Theorem
 the product of the means must equal the product of the extremes
 Corollary
 A theorem that follows quickly, easily, and directly from another theorem.
 Corollary 71

If \begin{align*}a, b, c,\end{align*}
a,b,c, and \begin{align*}d\end{align*}d are nonzero and \begin{align*}\frac{a}{b}=\frac{c}{d}\end{align*}ab=cd , then \begin{align*}\frac{a}{c}=\frac{b}{d}\end{align*}ac=bd .
 Corollary 72

Corollary 72 If \begin{align*}a, b, c,\end{align*}
a,b,c, and \begin{align*}d\end{align*}d are nonzero and \begin{align*}\frac{a}{b}=\frac{c}{d}\end{align*}ab=cd , then \begin{align*}\frac{d}{b}=\frac{c}{a}\end{align*}db=ca .
 Corollary 73

Corollary 73 If \begin{align*}a, b, c,\end{align*}
a,b,c, and \begin{align*}d\end{align*}d are nonzero and \begin{align*}\frac{a}{b}=\frac{c}{d}\end{align*}ab=cd , then \begin{align*}\frac{b}{a}=\frac{d}{c}\end{align*}ba=dc .
 Corollary 74

Corollary 74 If \begin{align*}a, b, c,\end{align*}
a,b,c, and \begin{align*}d\end{align*}d are nonzero and \begin{align*}\frac{a}{b}=\frac{c}{d}\end{align*}ab=cd , then \begin{align*}\frac{a+b}{b}=\frac{c+d}{d}\end{align*}a+bb=c+dd .
 Corollary 75

Corollary 75 If \begin{align*}a, b, c,\end{align*}
a,b,c, and \begin{align*}d\end{align*}d are nonzero and \begin{align*}\frac{a}{b}=\frac{c}{d}\end{align*}ab=cd , then \begin{align*}\frac{ab}{b}=\frac{cd}{d}\end{align*}a−bb=c−dd .
 Similar Polygons
 Two polygons with the same shape, but not the same size.
 Scale Factor
 In similar polygons, the ratio of one side of a polygon to the corresponding side of the other.
 Theorem 72
 The ratio of the perimeters of two similar polygons is the same as the ratio of the sides.
 AA Similarity Postulate
 If two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar.
 Indirect Measurement
 An application of similar triangles is to measure lengths indirectly.
 SSS Similarity Theorem
 If the corresponding sides of two triangles are proportional, then the two triangles are similar.
 SAS Similarity Theorem
 If two sides in one triangle are proportional to two sides in another triangle and the included angle in the first triangle is congruent to the included angle in the second, then the two triangles are similar.
 Triangle Proportionality Theorem
 If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.
 Triangle Proportionality Theorem Converse
 If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
 Theorem 77
 If three parallel lines are cut by two transversals, then they divide the transversals proportionally.
 Theorem 78
 If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides.
 Transformation
 An operation that moves, flips, or changes a figure to create a new figure.
 Rigid Transformation
 Transformations that preserve size are rigid
 Nonrigid Transformation
 Transformations that preserve size are rigid and ones that do not are nonrigid.
 Dilation
 A nonrigid transformation that preserves shape but not size.
 SelfSimilar
 When one part of an object can be enlarged (or shrunk) to look like the whole object.
 Fractal
 A fractal is another selfsimilar object that is repeated at successively smaller scales.
Review Questions
 Solve the following proportions.

\begin{align*}\frac{x+3}{3}=\frac{10}{2}\end{align*}
x+33=102 
\begin{align*}\frac{8}{5}=\frac{2x1}{x+3}\end{align*}
85=2x−1x+3

\begin{align*}\frac{x+3}{3}=\frac{10}{2}\end{align*}
 The extended ratio of the angle in a triangle are 5:6:7. What is the measure of each angle?
 Rewrite 15 quarts in terms of gallons.
Determine if the following pairs of polygons are similar. If it is two triangles, write why they are similar.
 Draw a dilation of \begin{align*}A(7, 2), B(4, 9),\end{align*}
A(7,2),B(4,9), and \begin{align*}C(1, 4)\end{align*}C(−1,4) with \begin{align*}k=\frac{3}{2}\end{align*}k=32 .
Algebra Connection Find the value of the missing variable(s).
Texas Instruments Resources
In the CK12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9692.
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Date Created:
Feb 22, 2012
Last Modified:
Feb 03, 2016
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