Question

Area of Triangles Quick Check 1 of 51 of 5 Items Question Use the image to answer the 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 9 inches2 9 inches squared 19 inches2 19 inches squared 82 inches2 82 inches squared 41 inches2 41 inches squared Skip to navigation page 14 of 14

Answer 1

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

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

Where:
- \( a = 7 \) inches
- \( b = 12 \) inches
- \( C = 77^\circ \)

Now, we can substitute the values into the formula:

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

Calculating the sine of \( 77^\circ \):

\[
\sin(77^\circ) \approx 0.9749
\]

Now we can calculate the area:

\[
\text{Area} = \frac{1}{2} \times 7 \times 12 \times 0.9749
\]
\[
= \frac{1}{2} \times 84 \times 0.9749
\]
\[
= 42 \times 0.9749 \approx 40.9998
\]

Rounding this to the nearest inch gives:

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

Thus, the correct answer is \( 41 \) inches\(^2\).