Question

course activity: pythagorean triples
question 3
jannette says \\( \triangle a b c \sim \triangle d e f \\) because \\( \triangle a b c \\)s sides form a pythagorean triple and \\( \triangle d e f \\)s side lengths are multiples of \\( \triangle a b c \\)s side lengths. is she correct? explain your
reasoning.

Explanation:

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

Use the Pythagorean theorem $a^{2}+b^{2}=c^{2}$. For $\triangle ABC$ with $a = 20$, $b = 21$, and $c$ (hypotenuse) to be checked.
$$20^{2}+21^{2}=400 + 441=841$$
$$29^{2}=841$$
So, $\triangle ABC$ is a right - triangle with sides $20$, $21$, $29$.

Step2: Check if $\triangle DEF$ is a right - triangle

For $\triangle DEF$ with $a = 40$, $b$ (let's find the missing side). Using the Pythagorean theorem:
Let the missing side of $\triangle DEF$ be $x$.
$$40^{2}+x^{2}=58^{2}$$
$$x^{2}=58^{2}-40^{2}=(58 + 40)(58 - 40)=98\times18 = 1764$$
$$x = 42$$

Step3: Check the ratio of sides

For $\triangle ABC$ sides $20$, $21$, $29$ and $\triangle DEF$ sides $40$, $42$, $58$.
The ratio of corresponding sides: $\frac{40}{20}=2$, $\frac{42}{21}=2$, $\frac{58}{29}=2$.

Since the ratios of the corresponding sides of $\triangle ABC$ and $\triangle DEF$ are equal and both are right - triangles (by Pythagorean theorem), $\triangle ABC\sim\triangle DEF$ by the Side - Side - Side (SSS) similarity criterion.

Answer:

Jannette is correct. $\triangle ABC\sim\triangle DEF$ because the ratios of their corresponding sides ($\frac{40}{20}=\frac{42}{21}=\frac{58}{29} = 2$) are equal and both are right - triangles (verified by $20^{2}+21^{2}=29^{2}$ and $40^{2}+42^{2}=58^{2}$). So, by SSS similarity, the triangles are similar.