Question

Use the trigonometry area formula to find the area (to the nearest inch) of △ABC △ 𝐴 𝐵 𝐶 , given a=7 inches 𝑎 = 7 inches , b=12 inches 𝑏 = 12 inches , and ∠C=77 degrees ∠ 𝐶 = 77 degrees .​(1 point) Responses 41 inches2 41 inches 2 41 inches squared 82 inches2 82 inches 2 82 inches squared 9 inches2 9 inches 2 9 inches squared 19 inches2

Answer 1

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

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

where:

• \( a \) and \( b \) are the lengths of two sides of the triangle,
• \( C \) is the angle between those two sides.

Given:

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

Now, let's substitute the values into the formula:

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

First, calculate \( \sin(77^\circ) \):

\[ \sin(77^\circ) \approx 0.97437 \]

Now substitute this value back into the area formula:

\[ \text{Area} \approx \frac{1}{2} \times 7 \times 12 \times 0.97437 \]

Calculating further:

\[ \text{Area} \approx \frac{1}{2} \times 84 \times 0.97437 \] \[ \text{Area} \approx 42 \times 0.97437 \] \[ \text{Area} \approx 40.99 \text{ square inches} \]

Rounding to the nearest inch, the area is approximately:

\[ \text{Area} \approx 41 \text{ square inches} \]

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