Question

the perimeter of triangle xyz is 24 units. what is the area of triangle xyz? round to the nearest tenth of a square unit. trigonometric area formula: area = \\(\frac{1}{2}ab\sin(c)\\) 14.7 square units 14.9 square units 15.0 square units 15.3 square units

Explanation:

Step1: Identify sides and angle

We know two sides: \( XY = 3 \), \( YZ = 11 \), and the included angle \( \angle XYZ = 102^\circ \)? Wait, no, the angle at \( X \) is \( 102^\circ \), and sides \( XY = 3 \), \( XZ \) we can find from perimeter. Perimeter is 24, so \( XY + YZ + XZ = 24 \), so \( 3 + 11 + XZ = 24 \), so \( XZ = 24 - 3 - 11 = 10 \). Wait, maybe I misread. Wait, the triangle has sides: \( XY = 3 \), \( YZ = 11 \), and \( XZ \) is calculated as \( 24 - 3 - 11 = 10 \). Then the angle at \( X \) is \( 102^\circ \), so the two sides forming the angle are \( XY = 3 \) and \( XZ = 10 \), and the included angle is \( 102^\circ \).

Step2: Apply the trigonometric area formula

The formula is \( \text{Area} = \frac{1}{2}ab\sin(C) \), where \( a = 3 \), \( b = 10 \), and \( C = 102^\circ \).

So, \( \text{Area} = \frac{1}{2} \times 3 \times 10 \times \sin(102^\circ) \)

First, calculate \( \sin(102^\circ) \). \( \sin(102^\circ) \approx \sin(180^\circ - 78^\circ) = \sin(78^\circ) \approx 0.9781 \)

Then, \( \frac{1}{2} \times 3 \times 10 = 15 \)

So, \( \text{Area} \approx 15 \times 0.9781 \approx 14.6715 \), which rounds to 14.7? Wait, no, wait maybe I messed up the sides. Wait, maybe the two sides are \( XY = 3 \) and \( YZ = 11 \), and the included angle? Wait, no, let's re-examine the diagram. The angle at \( X \) is \( 102^\circ \), between \( XY \) (length 3) and \( XZ \) (length we found as 10). Wait, maybe the correct sides are \( XY = 3 \), \( XZ = 10 \), angle at \( X \) is \( 102^\circ \). Then using the formula \( \frac{1}{2} \times 3 \times 10 \times \sin(102^\circ) \)

Wait, \( \frac{1}{2} \times 3 \times 10 = 15 \), then \( 15 \times \sin(102^\circ) \approx 15 \times 0.9781 \approx 14.6715 \), which is approximately 14.7? Wait, but let's check again. Wait, maybe the two sides are \( XY = 3 \) and \( YZ = 11 \), and the included angle? Wait, no, the angle at \( X \) is between \( XY \) and \( XZ \). Wait, maybe I made a mistake in calculating \( XZ \). Wait, perimeter is 24, so \( XY + YZ + XZ = 24 \), so \( 3 + 11 + XZ = 24 \), so \( XZ = 10 \). So sides: \( XY = 3 \), \( YZ = 11 \), \( XZ = 10 \). The angle at \( X \) is \( 102^\circ \), so between \( XY \) (3) and \( XZ \) (10). So the formula is \( \frac{1}{2} \times 3 \times 10 \times \sin(102^\circ) \). Let's compute that:

\( \frac{1}{2} \times 3 \times 10 = 15 \)

\( \sin(102^\circ) \approx 0.9781 \)

\( 15 \times 0.9781 \approx 14.6715 \), which rounds to 14.7. Wait, but let's check the options. The first option is 14.7. Wait, but maybe I messed up the sides. Wait, maybe the two sides are \( XY = 3 \) and \( YZ = 11 \), and the included angle? Wait, no, the angle is at \( X \), so between \( XY \) and \( XZ \). Alternatively, maybe the sides are \( XY = 3 \), \( XZ = 10 \), angle at \( X \) is \( 102^\circ \), so the area is \( \frac{1}{2} \times 3 \times 10 \times \sin(102^\circ) \approx 14.7 \).

Answer:

14.7 square units (Option: 14.7 square units)