Question

If the area of △ABC=134 square feet, m∠C = 41° , and b=24 feet, then what is the length of side a ? Round your answer to the nearest whole number.

(1 point)
Responses

16 feet
16 feet

9 feet
9 feet

17 feet
17 feet

15 feet

Answer 1

To find the length of side \( a \) in triangle \( \triangle ABC \), we can use the formula for the area of a triangle given by:

\[
\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)
\]

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

Given data:
- Area = \( 134 \) square feet
- \( \angle C = 41^\circ \)
- \( b = 24 \) feet

We need to find \( a \). Rearranging the area formula gives:

\[
a = \frac{2 \times \text{Area}}{b \times \sin(C)}
\]

Plugging in the values:

\[
a = \frac{2 \times 134}{24 \times \sin(41^\circ)}
\]

Now, we need to calculate \( \sin(41^\circ) \). Using a calculator:

\[
\sin(41^\circ) \approx 0.6561
\]

Now substituting this value back into the formula:

\[
a = \frac{2 \times 134}{24 \times 0.6561}
\]

Calculating the denominator:

\[
24 \times 0.6561 \approx 15.7824
\]

Then calculate \( a \):

\[
a = \frac{268}{15.7824} \approx 17.0
\]

Rounding to the nearest whole number, we find:

\[
a \approx 17 \text{ feet}
\]

Therefore, the length of side \( a \) is **17 feet**.