Question

what is the area of triangle pqr? round to the nearest tenth of a square unit. 70.5 square units 111.3 square units 185.4 square units 222.5 square units trigonometric area formula: area = \\(\frac{1}{2}ab\sin(c)\\) triangle image with q, p, r; qp=20, pr=12, angle at p is 68°, angle at q is 36°

Explanation:

Step1: Identify values for formula

We use the formula \( \text{Area} = \frac{1}{2}ab\sin(C) \). Here, \( a = 20 \), \( b = 12 \), and the included angle \( C \) (between sides \( a \) and \( b \)) needs to be found. First, find the angle at \( Q \): in triangle \( PQR \), angles sum to \( 180^\circ \). So \( \angle Q = 180^\circ - 36^\circ - 68^\circ = 76^\circ \)? Wait, no, wait. Wait, the sides: \( QP = 20 \), \( PR = 12 \), angle at \( P \) is \( 68^\circ \), angle at \( Q \) is \( 36^\circ \)? Wait, no, let's check the triangle. The formula is \( \frac{1}{2}ab\sin(C) \), where \( a \) and \( b \) are two sides, and \( C \) is the included angle. So if we take sides \( QP = 20 \) and \( PR = 12 \), the included angle is at \( P \), which is \( 68^\circ \)? Wait, no, wait the angle at \( Q \) is \( 36^\circ \), side \( QP = 20 \), side \( PR = 12 \), angle at \( P \) is \( 68^\circ \). Wait, let's confirm the included angle. The formula \( \frac{1}{2}ab\sin(C) \) requires \( C \) to be the angle between sides \( a \) and \( b \). So if we take \( a = 20 \) (QP), \( b = 12 \) (PR), then the included angle is at \( P \), which is \( 68^\circ \). Wait, no, maybe I misread. Wait, the triangle: points \( Q \), \( P \), \( R \). \( QP = 20 \), \( PR = 12 \), angle at \( P \) is \( 68^\circ \), angle at \( Q \) is \( 36^\circ \). So using the formula \( \text{Area} = \frac{1}{2} \times QP \times PR \times \sin(\angle P) \). So \( a = 20 \), \( b = 12 \), \( C = 68^\circ \).

Step2: Plug into formula

Calculate \( \frac{1}{2} \times 20 \times 12 \times \sin(68^\circ) \). First, \( \frac{1}{2} \times 20 \times 12 = 120 \). Then, \( \sin(68^\circ) \approx 0.9272 \). Then, \( 120 \times 0.9272 \approx 111.264 \), which rounds to \( 111.3 \).

Answer:

111.3 square units