Question

unit activity: right triangles and trigonometry
part a
is △abc a right triangle? explain your reasoning.
yes its is a right triangle. the side lengths satisfy the pythagorean theorem confirming it is a right triangle.
space used(includes formatting): 119 / 30000
part b
how can you find the length of \\(\overline{rt}\\) using similarity? explain your reasoning.

Explanation:

Step1: Verify Pythagorean theorem for △ABC

Check if $5^2 + 12^2 = 13^2$

$$\begin{align} 5^2 + 12^2 &= 25 + 144 = 169 \\ 13^2 &= 169 \end{align}$$

Step2: Confirm right triangle status

Since $5^2 + 12^2 = 13^2$, △ABC is right.

Step3: Prove similarity of the two triangles

First, find hypotenuse of △RST: $\sqrt{24^2 + 10^2} = \sqrt{576+100} = \sqrt{676}=26$.
Ratio of sides: $\frac{5}{10}=\frac{12}{24}=\frac{13}{26}=\frac{1}{2}$, so △ABC ~ △STR.

Step4: Use similarity to find RT

Set up proportion: $\frac{RT}{AC} = \frac{RS}{BC}$
$\frac{RT}{13} = \frac{24}{5}$

Answer:

Part A: Yes, $\triangle ABC$ is a right triangle. Its side lengths satisfy the Pythagorean theorem: $5^2 + 12^2 = 13^2$, so it is a right triangle with the right angle at $B$.
Part B: First, confirm $\triangle ABC \sim \triangle STR$ (all corresponding sides are in the ratio $\frac{1}{2}$: $\frac{5}{10}=\frac{12}{24}=\frac{13}{26}$). Then use the proportion of corresponding sides: $\frac{RT}{AC} = \frac{RS}{BC}$. Substituting values: $\frac{RT}{13} = \frac{24}{5}$, so $RT = \frac{24 \times 13}{5} = 62.4$ (or $\frac{312}{5}$).