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.
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

17 feet
17 feet

16 feet
16 feet

9 feet
9 feet

15 feet

Answer 1

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

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

Where:
- \( a \) is the length of side \( BC \)
- \( b \) is the length of side \( AC \)
- \( C \) is the angle at vertex \( C \)

We know:
- The area of triangle \( ABC = 134 \) square feet
- \( m\angle C = 41^\circ \)
- \( b = 24 \) feet

Substituting the known values into the area formula:

\[
134 = \frac{1}{2} \cdot a \cdot 24 \cdot \sin(41^\circ)
\]

Now we can solve for \( a \):

1. Calculate \( \sin(41^\circ) \):

Using a calculator, \( \sin(41^\circ) \approx 0.6561 \).

2. Substitute \( \sin(41^\circ) \) into the area formula:

\[
134 = \frac{1}{2} \cdot a \cdot 24 \cdot 0.6561
\]

3. Simplify:

\[
134 = 12a \cdot 0.6561
\]

4. Solve for \( a \):

\[
134 = 7.8732a
\]

\[
a = \frac{134}{7.8732} \approx 17.0
\]

Rounding to the nearest whole number, we get:

\[
\boxed{17}
\]

Thus, the length of side \( a \) is approximately 17 feet.