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Study Guide and Review
Choose the letter of the word or phrase that best completes each statement.
a. ratio
b. proportion
c. means
d. extremes
e. similar
f. scale factor
g. AA Similarity Post
h. SSS Similarity Theorem
i. SAS Similarity Theorem
j. midsegment
k. dilation
l. enlargement
m. reduction
1. A(n)
of a triangle has endpoints that are the midpoints of two sides of the triangle.
SOLUTION: j; midsegment
2. A(n)
is a comparison of two quantities using division.
SOLUTION: a; ratio
3. If
A
X and
C
Z, then
by the .
SOLUTION: g; AA Similarity Post. 4. A(n)
is an example of a similarity transformation.
SOLUTION: k; dilation
5. If
then a and d are the
SOLUTION: d; extremes
6. The ratio of the lengths of two corresponding sides of two similar polygons is the
SOLUTION: f; scale factor
7. A(n)
is an equation stating that two ratios are equivalent.
SOLUTION: b; proportion
8. A dilation with a scale factor of
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SOLUTION: m; reduction
will result in a(n)
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7. A(n)
is an equation stating that two ratios are equivalent.
SOLUTION: Study
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b; proportion
8. A dilation with a scale factor of
will result in a(n)
SOLUTION: m; reduction
Solve each proportion.
9. SOLUTION: Cross multiply.
Solve for x.
10. SOLUTION: Cross multiply.
Solve for x.
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Cross multiply.
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11. SOLUTION: Cross multiply.
Solve for x.
12. SOLUTION: Cross multiply.
Solve for x.
13. The ratio of the lengths of the three sides of a triangle is 5:8:10. If its perimeter is 276 inches, find the length of the
longest side of the triangle.
SOLUTION: Just as the ratio
or 5:8 is equivalent to
or 5x:8x , the extended ratio can be written as 5x:8x:10x. The perimeter is 276 inches, so the sum of the lengths of the sides is 276. Solve for x.
.
So the measures of the three sides are 5(12) or 60, 8(12) or 96, and 10(12) or 120. Thus the length of the longest side
is 120 inches.
A board
that
14. CARPENTRY
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the lengths of the two pieces.
SOLUTION: is 12 feet long must be cut into two pieces that have lengths in a ratio of 3 to 2. Find
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the measures
of the three sides are 5(12) or 60, 8(12) or 96, and 10(12) or 120. Thus the length of the longest side
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is 120 inches.
14. CARPENTRY A board that is 12 feet long must be cut into two pieces that have lengths in a ratio of 3 to 2. Find
the lengths of the two pieces.
SOLUTION: Let x and 12 – x be the lengths of the two pieces. Form a proportion.
Solve for x.
So, the length of the other piece is (12 – 7.2) ft or 4.8 ft.
Thus the length of the two pieces are 7.2 ft and 4.8 ft.
Determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If
not, explain your reasoning.
15. SOLUTION: Step 1: Compare corresponding angles: All corresponding angles are congruent. Step 2: Compare corresponding sides:
Since
, the figures are not similar. No, the polygons are not similar because the corresponding sides are not proportional.
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Since
, the figures are not similar. Study
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No,
the polygons
are not similar because the corresponding sides are not proportional.
16. SOLUTION: Step 1: Compare corresponding angles: Since all of the angles in the polygons are right angles, they are all congruent to each other. Therefore, corresponding angles are congruent. Step 2: Compare corresponding sides:
Since
, the corresponding sides of the polygons have the same scale factor, which is . Yes, the rectangles are similar because all of the corresponding angles are congruent and the corresponding sides
are proportional in a 3:2 ratio.
17. The two triangles in the figure below are similar. Find the value of x.
SOLUTION: If two triangles are similar, then their corresponding sides are proportional.
Form a proportion and solve for x. 18. PHOTOS IF the dimensions of a photo are 2 inches by 3 inches and the dimensions of a poster are 8 inches by 12
inches, are the photo and poster similar? Explan.
SOLUTION: Yes; the ratios are the same. eSolutions Manual - Powered by Cognero
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Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
Study Guide and Review
18. PHOTOS IF the dimensions of a photo are 2 inches by 3 inches and the dimensions of a poster are 8 inches by 12
inches, are the photo and poster similar? Explan.
SOLUTION: Yes; the ratios are the same. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
19. SOLUTION: By the Reflexive Property,
.
So,
by SAS Similarity.
20. SOLUTION: .
So,
by SSS Similarity.
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Study
So,
by SSS Similarity.
21. SOLUTION: No, the triangles are not similar because not all corresponding angles are congruent.
22. SOLUTION: Since
Yes,
, then congruent alternate interior angles are formed. Therefore,
. by the AA Similarity Post.
23. TREES To estimate the height of a tree, Dave stands in the shadow of the tree so that his shadow and the tree’s
shadow end at the same point. Dave is 6 feet 4 inches tall and his shadow is 15 feet long. If he is standing 66 feet
away from the tree, what is the height of the tree?
SOLUTION: Make a sketch of the situation. 4 feet 6 inches is equivalent to 4.5 feet.
In shadow problems, you can assume that the angles formed by the Sun’s rays with any two objects are congruent
and that the two objects form the sides of two right triangles.
Since two pairs of angles are congruent, the right triangles are similar by the AA Similarity Postulate. So, the following proportion can be written.
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Convert all measures to inches:
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SOLUTION: Since
, then congruent alternate interior angles are formed. Therefore,
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Yes,
by the AA Similarity Post.
. 23. TREES To estimate the height of a tree, Dave stands in the shadow of the tree so that his shadow and the tree’s
shadow end at the same point. Dave is 6 feet 4 inches tall and his shadow is 15 feet long. If he is standing 66 feet
away from the tree, what is the height of the tree?
SOLUTION: Make a sketch of the situation. 4 feet 6 inches is equivalent to 4.5 feet.
In shadow problems, you can assume that the angles formed by the Sun’s rays with any two objects are congruent
and that the two objects form the sides of two right triangles.
Since two pairs of angles are congruent, the right triangles are similar by the AA Similarity Postulate. So, the following proportion can be written.
Convert all measures to inches:
81 ft = 972 inches
6 feet 4 inches = 76 inches
15 feet = 180 inches.
Let x be the tree’s height. Substitute and solve for x:
So, the tree is 410.4 inches or 34.2 feet tall.
Find x.
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Solve for x.
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So, the tree is 410.4 inches or 34.2 feet tall.
Find x.
24. SOLUTION: Solve for x.
25. SOLUTION: Solve for x.
26. STREETS Find the distance along Broadway between 37th St. and 36th St.
SOLUTION: Assuming that 38th street, 37th street and 36th street are parallel to each other, we can set up a proportion using the
triangle proportionality Theorem:
.
Solve for x.
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Solve for x.
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26. STREETS Find the distance along Broadway between 37th St. and 36th St.
SOLUTION: Assuming that 38th street, 37th street and 36th street are parallel to each other, we can set up a proportion using the
triangle proportionality Theorem:
.
Solve for x.
The distance between 37th St. and 36th St is 220 ft.
Find the value of each variable.
27. SOLUTION: An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of
the other two sides.
28. SOLUTION: An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of
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28. SOLUTION: An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of
the other two sides.
29. MAPS The scale given on a map of the state of Missouri indicates that 3 inches represents 50 miles. The cities of
St. Louis, Springfield , and Kansas City form a triangle. If the measurements of the lengths of the sides of this
triangle on the map are 15 inches, 10 inches, and 13 inches, find the perimeter of the actual triangle formed by these
cities to the nearest mile.
SOLUTION: Perimeter of the triangle on the map = 15 + 10 + 13 = 38 inches
Assume that 38 inches represents x miles.
Form a proportion and solve for x. Therefore, the perimeter of the actual triangle is about 633 miles.
Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of
the dilation.
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SOLUTION: B is smaller than A, so the dilation is a reduction. Page 11
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Therefore, the perimeter of the actual triangle is about 633 miles.
Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of
the dilation.
30. SOLUTION: B is smaller than A, so the dilation is a reduction. The distance between the vertices at (0, 0) and (6,0) for A is 6 and between the vertices at (0, 0) and (2,0) for B is
2. So the scale factor is
or
.
31. SOLUTION: B is larger than A, so the dilation is an enlargement.
The distance between the vertices at (0, 1) and (2, 1) for A is 2.
The distance between the vertices at (3, 1) and (7, 1) for B is 4.
So the scale factor is
.
32. GRAPHIC DESIGN Jamie wants to use a photocopier to enlarge her design for the Honors Program at her school.
She sets the copier to 250%. If the original drawing was 6 inches by 9 inches, find the dimensions of the
enlargement.
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Therefore, the dimensions of the enlargement is 15 in. by 22.5 in..
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33. BUILDING PLANS In a scale drawing of a school’s floor plan, 6 inches represents 100 feet. If the distance from
The distance between the vertices at (3, 1) and (7, 1) for B is 4.
Study Guide and Review
So the scale factor is
.
32. GRAPHIC DESIGN Jamie wants to use a photocopier to enlarge her design for the Honors Program at her school.
She sets the copier to 250%. If the original drawing was 6 inches by 9 inches, find the dimensions of the
enlargement.
SOLUTION: Therefore, the dimensions of the enlargement is 15 in. by 22.5 in..
33. BUILDING PLANS In a scale drawing of a school’s floor plan, 6 inches represents 100 feet. If the distance from
one end of the main hallway to the other is 175 feet, find the corresponding length in the scale drawing.
SOLUTION: Assume that x inches represents 175 feet. Form a proportion and solve for x. The corresponding length in the scale drawing is 10.5 inches.
34. MODEL TRAINS A popular scale for model trains is the 1: 48 scale. If the actual train car had a length of 72 feet,
find the corresponding length of the model in inches.
SOLUTION: Let x be the unknown length of the model train.
Form a proportion and solve for x: .
The corresponding length of the model is 1.5 feet or 18 inches.
35. MAPS A map of the eastern United States has a scale where 3 inches = 25 miles. If the distance on the map
between Columbia, South Carolina, and Charlotte, North Carolina, is 11.5 inches what is the actual distance between
the cities?
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SOLUTION: Let x represent the actual distance between the cities. Page 13
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The corresponding length of the model is 1.5 feet or 18 inches.
35. MAPS A map of the eastern United States has a scale where 3 inches = 25 miles. If the distance on the map
between Columbia, South Carolina, and Charlotte, North Carolina, is 11.5 inches what is the actual distance between
the cities?
SOLUTION: Let x represent the actual distance between the cities. Form a proportion and solve for x: The actual distance between the cities is about 95.8 miles.
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