Question 408·Medium·Math

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In right triangle $PQR$ , angle $Q$ is the right angle. If $PQ = 9$ and $PR = 15$ , what is the value of $\sin(P)$ ?

A right triangle with two sides given and a sine value requested calls for naming the sides from the specified angle. First find the hypotenuse across from the right angle, then the leg opposite the requested angle. If the lengths fit a scaled $3$ - $4$ - $5$ triangle, use that pattern to find the missing leg. Don't use the adjacent leg for sine.

Hints

Hint 1
The hypotenuse is always across from the right angle. Once you've found it, look from angle $P$ : which of the remaining sides touches $P$ , and which lies across from $P$ ?
Hint 2
A Pythagorean triple is a set of whole-number side lengths that forms a right triangle; $3$ , $4$ , and $5$ are one example. Compare the known leg and hypotenuse with $3$ and $5$ . What multiplier connects each pair?

Step-by-step

Use a scaled right triangle
Step 1
Name the sides from angle P
No Desmos needed. These lengths fit a scaled $3$ - $4$ - $5$ right triangle. The right angle is at $Q$ , so $PR$ is the hypotenuse, the side across from it. The sides meeting at $Q$ are legs. From angle $P$ , $PQ$ is the leg that touches $P$ , while $QR$ is the opposite leg across from $P$ .
Step 2
Find the opposite leg
The $3$ - $4$ - $5$ right triangle has legs $3$ and $4$ and hypotenuse $5$ . Here $PQ=9=3\times3$ and $PR=15=3\times5$ , so both known sides have the same scale factor, $3$ . Scale the remaining leg by $3$ :
$QR=3\times4=12.$
Step 3
Write the sine ratio
For angle $P$ , sine is opposite leg over hypotenuse. Use $QR$ on top and $PR$ on the bottom:
$\sin(P)=\frac{QR}{PR}=\frac{12}{15}.$
Divide the numerator and denominator by $3$ :
$\sin(P)=\frac45.$
That's the requested value of $\sin(P)$ . Choice B.

Question 408·Medium·Math

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