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7.8: Chapter 7 Review

Difficulty Level: At Grade Created by: CK-12

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.
Cross-Multiplication 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 7-1
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 7-2
Corollary 7-2 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 7-3
Corollary 7-3 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 7-4
Corollary 7-4 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 7-5
Corollary 7-5 If \begin{align*}a, b, c,\end{align*}a,b,c, and \begin{align*}d\end{align*} are nonzero and \begin{align*}\frac{a}{b}=\frac{c}{d}\end{align*}, then \begin{align*}\frac{a-b}{b}=\frac{c-d}{d}\end{align*}.
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 7-2
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 7-7
If three parallel lines are cut by two transversals, then they divide the transversals proportionally.
Theorem 7-8
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
Non-rigid Transformation
Transformations that preserve size are rigid and ones that do not are non-rigid.
Dilation
A non-rigid transformation that preserves shape but not size.
Self-Similar
When one part of an object can be enlarged (or shrunk) to look like the whole object.
Fractal
A fractal is another self-similar object that is repeated at successively smaller scales.

Review Questions

  1. Solve the following proportions.
    1. \begin{align*}\frac{x+3}{3}=\frac{10}{2}\end{align*}
    2. \begin{align*}\frac{8}{5}=\frac{2x-1}{x+3}\end{align*}
  2. The extended ratio of the angle in a triangle are 5:6:7. What is the measure of each angle?
  3. 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.

  1. Draw a dilation of \begin{align*}A(7, 2), B(4, 9),\end{align*} and \begin{align*}C(-1, 4)\end{align*} with \begin{align*}k=\frac{3}{2}\end{align*}.

Algebra Connection Find the value of the missing variable(s).

Texas Instruments Resources

In the CK-12 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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