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

19 inches2
19 inches squared

9 inches2
9 inches squared

82 inches2
82 inches squared

41 inches2

Answer 1

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

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

where:
- \( a \) is the length of side \( a \),
- \( b \) is the length of side \( b \),
- \( C \) is the angle opposite side \( c \).

In this case:
- \( a = 7 \) inches,
- \( b = 12 \) inches,
- \( \angle C = 77^\circ \).

First, we need to calculate \( \sin(77^\circ) \). Using a calculator, we find:

\[
\sin(77^\circ) \approx 0.974
\]

Now, we can plug in the values into the area formula:

\[
\text{Area} = \frac{1}{2} \times 7 \times 12 \times \sin(77^\circ)
\]

Calculating this step by step:

1. Calculate \( \frac{1}{2} \times 7 \times 12 \):

\[
\frac{1}{2} \times 7 \times 12 = \frac{84}{2} = 42
\]

2. Now, multiply by \( \sin(77^\circ) \):

\[
\text{Area} = 42 \times 0.974 \approx 40.908
\]

Rounding to the nearest inch, the area is:

\[
\text{Area} \approx 41 \text{ inches}^2
\]

Therefore, the correct answer is:

**41 inches²**.