Question

use one of the triangles to approximate pq in the triangle below.

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( PQR \) with right angle at \( Q \), we know the hypotenuse \( PR = 7 \) (wait, no, looking at the diagram, \( PR \) is 7? Wait, the angle at \( R \) is \( 50^\circ \), and we want to find \( PQ \). In a right triangle, \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). Here, \( \theta = 50^\circ \), opposite side to \( \theta \) is \( PQ \), and hypotenuse is \( PR \). Wait, but the diagram shows \( PR \) as 7? Wait, maybe I misread. Wait, the problem says "use one of the triangles" – maybe it's a right triangle, so \( \sin(50^\circ)=\frac{PQ}{PR} \). Wait, if \( PR = 7 \), then \( PQ = PR \times \sin(50^\circ) \).

Step2: Calculate \( \sin(50^\circ) \)

We know that \( \sin(50^\circ) \approx 0.7660 \).

Step3: Compute \( PQ \)

\( PQ = 7 \times \sin(50^\circ) \approx 7 \times 0.7660 = 5.362 \). Wait, but maybe the hypotenuse is 10? Wait, the top has 10, maybe I misread the diagram. Wait, the triangle: \( R \) has \( 50^\circ \), right angle at \( Q \), so \( PR \) is the hypotenuse? Wait, maybe the length of \( PR \) is 7? Or is it 10? Wait, the text says "use one of the triangles" – maybe the triangle has hypotenuse 7? Wait, no, the diagram: \( PR \) is labeled 7, \( RQ \) is... Wait, no, let's re-express. In right triangle \( PQR \), \( \angle Q = 90^\circ \), \( \angle R = 50^\circ \), side \( PR = 7 \) (hypotenuse). Then \( PQ \) is opposite \( \angle R \), so \( \sin(50^\circ) = \frac{PQ}{PR} \), so \( PQ = PR \times \sin(50^\circ) \).
Calculating: \( \sin(50^\circ) \approx 0.7660 \), so \( PQ \approx 7 \times 0.7660 \approx 5.36 \). Alternatively, if the hypotenuse is 10, but the diagram shows 7 on \( PR \). Wait, maybe the 10 is a typo, or maybe I misread. Wait, the problem says "use one of the triangles" – perhaps the triangle has \( PR = 7 \), angle at \( R = 50^\circ \), right angle at \( Q \). So using sine: \( \sin(50^\circ) = \frac{PQ}{PR} \), so \( PQ = 7 \times \sin(50^\circ) \approx 7 \times 0.7660 \approx 5.36 \).

Answer:

Approximately \( 5.36 \) (or more accurately, using calculator: \( \sin(50^\circ) \approx 0.7660 \), so \( 7 \times 0.7660 = 5.362 \), so approximately \( 5.36 \) or \( 5.4 \) depending on rounding).