12 multiple choice 1 point in the right triangle shown, what is the length of $ \overline { p q } $ ? 13 multiple choice 1 point a man who is 6 feet tall casts a shadow 8 feet long. his son casts a shadow 4 feet long. how tall is the son?

Question

Accepted Answer

# Explanation:
## Step1: Apply Pythagorean theorem
For the right - triangle $ \triangle PQR$ (assuming the right - angle at $Q$), by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 13$ (hypotenuse) and $b = 12$ (one leg). Let $a=\overline{PQ}$. Then $a^{2}+12^{2}=13^{2}$.
## Step2: Solve for $a$
$$
\begin{align*}
a^{2}&=13^{2}-12^{2}\
a^{2}&=(13 + 12)(13 - 12)\
a^{2}&=25\times1\
a^{2}&=25\
a& = 5
\end{align*}
$$

## Step3: Set up proportion for son's height
Let the man's height $h_{1}=6$ ft, man's shadow $s_{1}=8$ ft, son's shadow $s_{2}=4$ ft, and son's height $h_{2}$. Since the triangles (formed by the man and his shadow, son and his shadow) are similar, $\frac{h_{1}}{s_{1}}=\frac{h_{2}}{s_{2}}$.
## Step4: Solve for $h_{2}$
$$
\begin{align*}
\frac{6}{8}&=\frac{h_{2}}{4}\
8h_{2}&=6\times4\
8h_{2}&=24\
h_{2}&=3
\end{align*}
$$

# Answer:
12. $5\mathrm{cm}$
13. $3\mathrm{ft}$

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QUESTION IMAGE

Question

Explanation:

Step1: Apply Pythagorean theorem

Step2: Solve for \(a\)

Step3: Set up proportion for son's height

Step4: Solve for \(h_{2}\)

Answer: