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

15 feet
15 feet

9 feet
9 feet

17 feet
17 feet

16 feet

Answer 1

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

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

Where \( a \) and \( b \) are the lengths of two sides and \( C \) is the angle between them.

From the information provided:
- Area = 134 square feet
- \( b = 24 \) feet
- \( m\angle C = 41^\circ \)

We can substitute these values into the area formula and solve for \( a \):

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

First, we calculate \( \sin(41^\circ) \):

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

Now we substitute this value into the equation:

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

Simplifying we get:

\[
134 = 12a \cdot 0.6561
\]

\[
134 = 7.8732a
\]

Now, to find \( a \):

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

Rounding to the nearest whole number, we find:

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

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