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Study Guide and Review
Solve each proportion.
9. SOLUTION: Cross multiply.
Solve for x.
10. SOLUTION: Cross multiply.
Solve for x.
11. SOLUTION: Cross multiply.
Solve for x.
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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.
Determine
whether
each pair
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not, explain your reasoning.
of figures is similar. If so, write the similarity statement and scale factor.Page
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SoGuide
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
Study
and Review
is 120 inches.
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.
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.
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17. The two triangles in the figure below are similar. Find the value of x.
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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. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
19. SOLUTION: By the Reflexive Property,
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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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No, the triangles are not similar because not all corresponding angles are congruent.
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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.
24. SOLUTION: eSolutions Manual - Powered by Cognero
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.
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