question 3\njannette says △abc ~ △def because △abcs sides form a pythagorean triple and △defs side lengths…

Explanation:

Step1: Check if $\triangle ABC$ is a right - triangle

In $\triangle ABC$, by the Pythagorean theorem, we check if $AB^{2}+BC^{2}=AC^{2}$. Given $AB = 20$, $BC=21$, then $AB^{2}=20^{2}=400$, $BC^{2}=21^{2}=441$, and $AB^{2}+BC^{2}=400 + 441=841$. Also, if $AC$ is the hypotenuse, $AC=\sqrt{841}=29$. So, $\triangle ABC$ is a right - triangle with side lengths $20,21,29$.

Step2: Check the ratio of side - lengths of $\triangle ABC$ and $\triangle DEF$

The side - lengths of $\triangle ABC$ are $20,21,29$ and the side - lengths of $\triangle DEF$ are $40,?,58$. Let's find the third side of $\triangle DEF$ using the Pythagorean theorem. Let the third side be $x$. Then $x=\sqrt{58^{2}-40^{2}}=\sqrt{(58 + 40)(58 - 40)}=\sqrt{98\times18}=\sqrt{1764}=42$. The ratio of corresponding sides: $\frac{40}{20}=2$, $\frac{42}{21}=2$, $\frac{58}{29}=2$.

Step3: Determine similarity

Since the ratios of the corresponding side - lengths of $\triangle ABC$ and $\triangle DEF$ are equal, by the Side - Side - Side (SSS) similarity criterion, $\triangle ABC\sim\triangle DEF$.

Answer:

Yes, she is correct. The ratios of the corresponding side - lengths of $\triangle ABC$ and $\triangle DEF$ are equal, so $\triangle ABC\sim\triangle DEF$ by the SSS similarity criterion.