Question

1. $\triangle abc$ is a right triangle with side lengths 7.5, 10, and 12.5. which sets of side lengths form triangles similar to $\triangle abc$? choose all that apply.

a. 1.5, 2, 2.5
b. 3, 4, 5
c. 6, 8, 9.5
d. 10, 12.5, 15
e. 15, 20, 25

1. $\triangle jkl$ and $\triangle mnp$ are similar triangles.

(image of two right triangles: $\triangle jkl$ with right angle at $l$, and $\triangle mnp$ with right angle at $p$)

a. write a proportion by using within - triangle ratios.

b. write a proportion by using between - triangle ratios.

Explanation:

Problem 1

Step1: Simplify △ABC side ratio

Divide each side by 2.5:
$\frac{7.5}{2.5}=3$, $\frac{10}{2.5}=4$, $\frac{12.5}{2.5}=5$. So the simplified ratio is $3:4:5$.

Step2: Check Option A

Divide each side by 0.5:
$\frac{1.5}{0.5}=3$, $\frac{2}{0.5}=4$, $\frac{2.5}{0.5}=5$. Ratio is $3:4:5$.

Step3: Check Option B

Ratio is already $3:4:5$.

Step4: Check Option C

Simplify: $\frac{6}{2}=3$, $\frac{8}{2}=4$, $\frac{9.5}{2}=4.75$. Ratio is $3:4:4.75$.

Step5: Check Option D

Simplify: $\frac{10}{2.5}=4$, $\frac{12.5}{2.5}=5$, $\frac{15}{2.5}=6$. Ratio is $4:5:6$.

Step6: Check Option E

Divide each side by 5:
$\frac{15}{5}=3$, $\frac{20}{5}=4$, $\frac{25}{5}=5$. Ratio is $3:4:5$.

Part a: Within-triangle ratios

For similar right triangles, the ratio of a leg to the hypotenuse (or other corresponding sides) within each triangle is equal.

Part b: Between-triangle ratios

Corresponding sides of similar triangles are proportional across the two triangles.

Answer:

A. 1.5, 2, 2.5
B. 3, 4, 5
E. 15, 20, 25

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Problem 2