Question

integrated 2 unit 3 similarity review
topic covered on this test:

• setting up proportions and solving for missing side lengths of similar figures.

similar figures

1. what is the same about similar figures and congruent figures?
2. what is different about similar figures and congruent figures?
3. solve for the value of the variable in the sets of similar figures below. show the proportions you finding the side length.

a. △cat∼△dog
b. △fgj∼△fhi
c.
d.
e. △abc∼△dbe
f.

Explanation:

Step1: Define similarity and congruence

Similar figures have equal - angle measures and proportional side - lengths. Congruent figures have equal - angle measures and equal side - lengths.

Step2: Answer question 1

The angle measures are the same. In both similar and congruent figures, corresponding angles are equal.

Step3: Answer question 2

The side - lengths are different. In similar figures, side - lengths are proportional but not necessarily equal. In congruent figures, side - lengths are equal.

Step4: Solve part a

Since $\triangle CAT\sim\triangle DOG$, we set up the proportion $\frac{CA}{DO}=\frac{AT}{OG}=\frac{CT}{DG}$. Using $\frac{CA}{DO}=\frac{AT}{OG}$, we have $\frac{6}{21.6}=\frac{2}{7.2}=\frac{5}{x}$. Cross - multiplying $\frac{6}{21.6}=\frac{5}{x}$, we get $6x = 5\times21.6$, so $x=\frac{5\times21.6}{6}=18$.

Step5: Solve part b

Since $\triangle FGJ\sim\triangle FHI$, we set up the proportion $\frac{FJ}{FI}=\frac{GJ}{HI}=\frac{FG}{FH}$. Using $\frac{FJ}{FI}=\frac{GJ}{HI}$, we have $\frac{10}{10 + 15}=\frac{8}{x}$, or $\frac{10}{25}=\frac{8}{x}$. Cross - multiplying gives $10x=8\times25$, so $x = 20$.

Step6: Solve part c

The two triangles are similar. We set up the proportion $\frac{7}{14}=\frac{9}{3x}$. Cross - multiplying gives $7\times3x=9\times14$, so $21x = 126$ and $x = 6$.

Step7: Solve part d

For the two rectangles (similar), we set up the proportion $\frac{12}{x}=\frac{20}{18}$. Cross - multiplying gives $20x=12\times18$, so $x=\frac{12\times18}{20}=10.8$.

Step8: Solve part e

Since $\triangle ABC\sim\triangle DBE$, we set up the proportion $\frac{AB}{DB}=\frac{BC}{BE}=\frac{AC}{DE}$. Using $\frac{AB}{DB}=\frac{BC}{BE}$, we have $\frac{10}{10 - 8}=\frac{x}{6}$, or $\frac{10}{2}=\frac{x}{6}$. Cross - multiplying gives $2x = 60$, so $x = 30$.

Step9: Solve part f

The two triangles are similar. We set up the proportion $\frac{6}{9}=\frac{x}{10}$. Cross - multiplying gives $9x=6\times10$, so $x=\frac{60}{9}=\frac{20}{3}\approx6.67$.

Answer:

1. The same: Corresponding angles are equal.
2. The different: In similar figures, side - lengths are proportional; in congruent figures, side - lengths are equal.
3. a. $x = 18$

b. $x = 20$
c. $x = 6$
d. $x = 10.8$
e. $x = 30$
f. $x=\frac{20}{3}$