In triangle PQR, the measure of angle P is 30^\circ, the measure of angle Q is 45^\circ, a… — ACT

In triangle $PQR$ , the measure of angle $P$ is $30^\circ$ , the measure of angle $Q$ is $45^\circ$ , and the length of side $QR$ is $10$ units. Which of the following expressions represents the length, in units, of side $PR$ ?

A
$10\sin(30^\circ)\sin(45^\circ)$
B
$\frac{10\sin(30^\circ)}{\sin(45^\circ)}$
C
$\frac{10\cos(45^\circ)}{\cos(30^\circ)}$
$\frac{10\sin(45^\circ)}{\sin(30^\circ)}$ Answer
E
$\frac{10\cos(30^\circ)}{\sin(45^\circ)}$

Answer

The length of side

PR

is represented by the expression

\frac{10\sin(45^\circ)}{\sin(30^\circ)}

The correct answer is derived by setting up the Law of Sines proportion:

\frac{PR}{\sin(45^\circ)} = \frac{10}{\sin(30^\circ)}

. Multiplying both sides by

\sin(45^\circ)

yields

PR = \frac{10\sin(45^\circ)}{\sin(30^\circ)}

Step-by-Step Solution

Identify the known values and corresponding angle-side pairs in triangle

PQR

Angle

P = 30^\circ

is opposite to side

QR = 10

, and angle

Q = 45^\circ

is opposite to side

PR

This allows us to set up the appropriate trigonometric relationship to solve for the unknown side.

Apply the Law of Sines to relate the ratios of the side lengths to the sines of their opposite angles.

\frac{PR}{\sin(Q)} = \frac{QR}{\sin(P)}

, which becomes

\frac{PR}{\sin(45^\circ)} = \frac{10}{\sin(30^\circ)}

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides.

Isolate the variable representing the length of side

PR

PR = \frac{10\sin(45^\circ)}{\sin(30^\circ)}

Multiply both sides of the equation by

\sin(45^\circ)

to solve for

PR

Key Concept

The Law of Sines relates the side lengths of a triangle to the sines of its angles:

\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

Estimated Time:1m 0s

Question

Answer

Step-by-Step Solution

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