Question

question 3
jannette says $\triangle abc \sim \triangle def$ because $\triangle abc$s sides form a pythagorean triple and $\triangle def$s side lengths are multiples of $\triangle abc$s side lengths. is she correct? explain your reasoning.

Explanation:

Step1: Find hypotenuse of △ABC

Use Pythagorean theorem: $AC = \sqrt{AB^2 + BC^2} = \sqrt{20^2 + 21^2}$
$AC = \sqrt{400 + 441} = \sqrt{841} = 29$

Step2: Find side EF of △DEF

Use Pythagorean theorem: $EF = \sqrt{DF^2 - DE^2} = \sqrt{58^2 - 40^2}$
$EF = \sqrt{3364 - 1600} = \sqrt{1764} = 42$

Step3: Check side ratios

Calculate ratios of corresponding sides:
$\frac{DE}{AB} = \frac{40}{20} = 2$, $\frac{EF}{BC} = \frac{42}{21} = 2$, $\frac{DF}{AC} = \frac{58}{29} = 2$

Step4: Verify similarity condition

All corresponding sides have equal ratios, and right angles are congruent, so triangles satisfy SSS similarity.

Answer:

Jannette is correct. The corresponding sides of $\triangle ABC$ and $\triangle DEF$ are in a consistent 1:2 ratio ($\frac{20}{40}=\frac{21}{42}=\frac{29}{58}=\frac{1}{2}$), and both are right triangles. By the Side-Side-Side (SSS) Similarity Criterion, $\triangle ABC \sim \triangle DEF$.