This handy Spectrum Math Grade 7 Answer Key Chapter 5 Lesson 5.1 Scale Drawings provides detailed answers for the workbook questions
Spectrum Math Grade 7 Chapter 5 Lesson 5.1 Scale Drawings Answers Key
Two triangles are similar if their corresponding (matching) angles are congruent (have the same measure) and the lengths of their corresponding sides are proportional.
These triangles are similar. All the sides are proportional.
\(\frac{A B}{D E}\) = \(\frac{12}{8}\) = \(\frac{3}{2}\)
\(\frac{B C}{E F}\) = \(\frac{12}{8}\) = \(\frac{3}{2}\)
\(\frac{A C}{D F}\) = \(\frac{9}{6}\) = \(\frac{3}{2}\)
The angle measures are congruent.
These triangles are not similar. The sides are not proportional. They do not all create the same ratio. The angle measures are not all congruent.
\(\frac{G H}{J K}\) = \(\frac{4}{3}\)
\(\frac{H I}{K L}\) = \(\frac{6}{5.86}\)
\(\frac{G I}{J L}\) = \(\frac{5}{5}\) = \(\frac{1}{1}\)
For each pair of triangles, check that their sides are proportional. Circle similar or not similar.
Question 1.
\(\frac{K M}{K^{\prime} M^{\prime}}\) = _____________ = ______________
\(\frac{K M}{K^{\prime} M^{\prime}}\) = _____________ = ______________ similar
\(\frac{M L}{M^{\prime} L^{\prime}}\) = _____________ = ______________ not similar
Answer:
\(\frac{K M}{K^{\prime} M^{\prime}}\) = \(\frac{2}{3}\)
\(\frac{K L}{K^{\prime} L^{\prime}}\) = \(\frac{2}{3}\) similar.
\(\frac{ML}{M^{\prime} L^{\prime}}\) = \(\frac{2}{3}\) similar.
Explanation:
Two triangles are similar if their corresponding (matching) angles are congruent (have the same measure) and the lengths of their corresponding sides are proportional.
These triangles are similar. All the sides are proportional.
\(\frac{K M}{K^{\prime} M^{\prime}}\) = \(\frac{24}{36}\) = \(\frac{2}{3}\)
\(\frac{K L}{K^{\prime} L^{\prime}}\) = \(\frac{28}{42}\) = \(\frac{2}{3}\)
\(\frac{M L}{M^{\prime} L^{\prime}}\) = \(\frac{36}{54}\) = \(\frac{2}{3}\)
The angle measures are congruent.
Question 2.
\(\frac{P N}{P^{\prime} N^{\prime}}\) = _____________ = ______________
\(\frac{P O}{P^{\prime} O^{\prime}}\) = _____________ = ______________ similar
\(\frac{N O}{N^{\prime} O^{\prime}}\) = _____________ = ______________ not similar
Answer:
\(\frac{P N}{P^{\prime} N^{\prime}}\) = \(\frac{3}{2}\)
\(\frac{P O}{P^{\prime} O^{\prime}}\) = \(\frac{3}{2}\) similar.
\(\frac{N O}{N^{\prime} O^{\prime}}\) = \(\frac{6}{5}\) not similar.
Explanation:
These triangles are not similar.
Because the sides are not proportional.
They do not all create the same ratio.
The angle measures are not all congruent.
\(\frac{P N}{P^{\prime} N^{\prime}}\) = \(\frac{18}{12}\) = \(\frac{3}{2}\)
\(\frac{P O}{P^{\prime} O^{\prime}}\) = \(\frac{12}{8}\) = \(\frac{3}{2}\)
\(\frac{N O}{N^{\prime} O^{\prime}}\) = \(\frac{12}{10}\) = \(\frac{6}{5}\)
Question 3.
\(\frac{Q S}{Q^{\prime} S^{\prime}}\) = _____________ = ______________
\(\frac{Q R}{Q^{\prime} R^{\prime}}\) = _____________ = ______________ similar
\(\frac{R S}{R^{\prime} S^{\prime}}\) = _____________ = ______________ not similar
Answer:
\(\frac{Q S}{Q^{\prime} S^{\prime}}\) = \(\frac{3}{4}\)
\(\frac{Q R}{Q^{\prime} R^{\prime}}\) = \(\frac{3}{4}\) similar.
\(\frac{R S}{R^{\prime} S^{\prime}}\) = \(\frac{3}{4}\) not similar.
Explanation:
Two triangles are similar if their corresponding (matching) angles are congruent (have the same measure) and the lengths of their corresponding sides are proportional.
These triangles are similar.
All the sides are proportional.
\(\frac{Q S}{Q^{\prime} S^{\prime}}\) = \(\frac{30}{40}\) = \(\frac{3}{4}\)
\(\frac{Q R}{Q^{\prime} R^{\prime}}\) = \(\frac{27}{36}\) = \(\frac{3}{4}\)
\(\frac{R S}{R^{\prime} S^{\prime}}\) = \(\frac{24}{32}\) = \(\frac{3}{4}\)
The angle measures are congruent.
When you know that two triangles are similar, you can use the ratio of the known lengths of the sides to figure the unknown length.
What is the length of EF?
\(\frac{A C}{D F}\) = \(\frac{B C}{E F}\) \(\frac{4}{6}\) = \(\frac{12}{n}\) Use a proportion.
4n = 72 n = 18 Cross multiply.
Find the length of the missing side for each pair of similar triangles. Label the side with its length.
Question 1.
a.
Answer:
21 ft.
Explanation:
\(\frac{AC}{A’C’}\) = \(\frac{AB}{A’B’}\)
\(\frac{12}{18}\) = \(\frac{14}{A’B’}\) Use a proportion.
12 x A’B’ = 14 x 18
A’B’ = \(\frac{18 × 14}{12}\)
A’B’ = 21 ft.
b.
Answer:
10 ft.
Explanation:
\(\frac{XZ}{X’Z’}\) = \(\frac{XY}{X’Y’}\)
\(\frac{12}{6}\) = \(\frac{20}{X’Y’}\) Use a proportion.
12 x X’Y’ = 14 x 18
X’Y’ = \(\frac{20 × 6}{12}\)
X’Y’ = 10 ft.
Question 2.
a.
Answer:
24 m.
Explanation:
\(\frac{PQ}{P’Q’}\) = \(\frac{PR}{P’R’}\)
\(\frac{28}{32}\) = \(\frac{21}{P’R’}\) Use a proportion.
28 x P’R’ = 21 x 32
P’R’ = \(\frac{21 × 32}{28}\)
P’R’ = 24 m.
b.
Answer:
25 in.
Explanation:
\(\frac{XZ}{X’Z’}\) = \(\frac{ZY}{Z’Y’}\)
\(\frac{12}{15}\) = \(\frac{20}{Z’Y’}\) Use a proportion.
12 x Z’Y’ = 20 x 15
Z’Y’ = \(\frac{20 × 15}{12}\)
Z’Y’ = 25 in.
Question 3.
a.
Answer:
15 cm.
Explanation:
\(\frac{AB}{A’B’}\) = \(\frac{BC}{B’C’}\)
\(\frac{18}{30}\) = \(\frac{9}{B’C’}\) Use a proportion.
18 x B’C’ = 30 x 9
B’C’ = \(\frac{30 × 9}{18}\)
B’C’ = 15 cm.
b.
Answer:
10 ft.
Explanation:
\(\frac{XY}{X’Y’}\) = \(\frac{ZY}{Z’Y’}\)
\(\frac{24}{15}\) = \(\frac{16}{Z’Y’}\) Use a proportion.
24 x Z’Y’ = 16 x 15
Z’Y’ = \(\frac{16 × 15}{24}\)
Z’Y’ = 10 ft.
Two figures are similar if their corresponding angles are congruent and the lengths of their corresponding sides are proportional. Write a ratio to determine if the sides are proportional.
In the following figures, the angle marks indicate which angles are congruent. Use the measures given for the lengths of the sides. Write ratios to determine if the sides are proportional. Then, write similar or not similar for each pair of figures.
Question 1.
a.
Answer:
\(\frac{AB}{XY} \) = \(\frac{1}{2} \)
\(\frac{AC}{XZ} \) = \(\frac{1}{2} \) similar.
\(\frac{BC}{YZ} \)= \(\frac{1}{2} \) similar.
Explanation:
Two triangles are similar if their corresponding (matching) angles are congruent (have the same measure) and the lengths of their corresponding sides are proportional.
These triangles are similar.
All the sides are proportional.
\(\frac{AB}{XY} \) = \(\frac{1}{2} \)
\(\frac{AC}{XZ} \) = \(\frac{1}{2} \)
\(\frac{BC}{YZ} \)= \(\frac{1}{2} \)
The angle measures are congruent.
b.
Answer:
\(\frac{AB}{WX} \) = \(\frac{2.3}{1.5} \)
\(\frac{AD}{WZ} \) = \(\frac{2.7}{2} \)
\(\frac{BC}{XY} \)= \(\frac{2}{1} \)
\(\frac{CD}{YZ} \)= \(\frac{1.5}{1} \)
Explanation:
Two parallelogram are not similar if their corresponding (matching) angles are not congruent (have the no same measure) and the lengths of their corresponding sides are not proportional.
These triangles are similar.
All the sides are proportional.
\(\frac{AB}{WX} \) = \(\frac{2.3}{1.5} \)
\(\frac{AD}{WZ} \) = \(\frac{2.7}{2} \)
\(\frac{BC}{XY} \)= \(\frac{1.5}{1} \)
\(\frac{CD}{YZ} \)= \(\frac{1.5}{1} \)
The angle measures are not congruent.
Question 2.
a.
Answer:
\(\frac{AB}{TU} \) = \(\frac{2}{3} \)
\(\frac{AE}{TX} \) = \(\frac{1}{2} \)
\(\frac{ED}{XW} \) = \(\frac{1}{1} \)
\(\frac{BC}{XY} \)= \(\frac{2}{1} \)
\(\frac{CD}{YZ} \)= \(\frac{1.5}{1} \)
Explanation:
Two parallelogram are not similar if their corresponding (matching) angles are not congruent (have the no same measure) and the lengths of their corresponding sides are not proportional.
These triangles are similar.
All the sides are not proportional.
\(\frac{AB}{TU} \) = \(\frac{2}{3} \)
\(\frac{AE}{TX} \) = \(\frac{1}{2} \)
\(\frac{ED}{XW} \) = \(\frac{1}{1} \)
\(\frac{BC}{XY} \)= \(\frac{2}{1} \)
\(\frac{CD}{YZ} \)= \(\frac{1.5}{1} \)
The angle measures are not congruent.
b.
Answer:
\(\frac{AB}{WX} \) = \(\frac{6}{3} \) = 2
\(\frac{BC}{XY} \)= \(\frac{12}{6} \) = 2
\(\frac{CD}{YZ} \)= \(\frac{10}{5} \) = 2
\(\frac{AD}{WZ} \) = \(\frac{5}{2.5} \) = 2
Explanation:
Two parallelogram are similar if their corresponding (matching) angles are congruent (have the no same measure) and the lengths of their corresponding sides are proportional.
These triangles are similar. All the sides are proportional.
\(\frac{AB}{WX} \) = \(\frac{6}{3} \) = 2
\(\frac{BC}{XY} \)= \(\frac{12}{6} \) = 2
\(\frac{CD}{YZ} \)= \(\frac{10}{5} \) = 2
\(\frac{AD}{WZ} \) = \(\frac{5}{2.5} \) = 2
The angle measures are congruent.