Question

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.
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**.