A triangle upper A upper B upper C. The edge upper A upper B is labeled c. The edge upper A upper C is labeled b. The edge upper B upper C is labeled a.

Question

Use the trigonometry area formula to find the area (to the nearest inch) of △ABC , given a=7 inches , b=12 inches , and ∠C=77 degrees .​

(1 point)
Responses

82 inches2
82 inches squared

9 inches2
9 inches squared

41 inches2
41 inches squared

19 inches2

Answer 1

To find the area of triangle \( \triangle ABC \) using the trigonometric area formula, we can use the formula:

\[ \text{Area} = \frac{1}{2}ab \sin(C) \]

where:

• \( a \) and \( b \) are the lengths of the sides adjacent to angle \( C \),
• \( C \) is the angle at vertex \( C \).

In this case:

• \( a = 7 \) inches,
• \( b = 12 \) inches,
• \( \angle C = 77^\circ \).

Now we can calculate the area:

1. Calculate \( \sin(77^\circ) \): \[ \sin(77^\circ) \approx 0.974 \quad (\text{using a calculator}) \]

2. Substitute the values into the area formula: \[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times \sin(77^\circ) \] \[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times 0.974 \]

3. Calculate the area: \[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times 0.974 \approx \frac{1}{2} \times 84 \times 0.974 \approx 42 \times 0.974 \approx 40.908 \]

4. Round this value to the nearest inch: \[ \text{Area} \approx 41 \text{ inches}^2 \]

Thus, the area of triangle \( \triangle ABC \) is approximately 41 inches².

The correct response is:

41 inches².