Question

Area of Triangles Quick Check 3 of 53 of 5 Items Question Use the image to answer the 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 9 feet 9 feet 16 feet 16 feet 15 feet 15 feet 17 feet 17 feet Skip to navigation page 14 of 14

Answer 1

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

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

Where:
- \( A \) is the area of the triangle,
- \( a \) and \( b \) are the lengths of two sides,
- \( C \) is the angle between those two sides.

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

Substituting the given values into the area formula:

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

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

Using a calculator, we find:
\[
\sin(41^\circ) \approx 0.6561
\]

Now substitute this value into the equation:

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

This simplifies to:

\[
134 = 12a \times 0.6561
\]

Calculating \( 12 \times 0.6561 \):

\[
12 \times 0.6561 \approx 7.9932
\]

Now, substitute back into the equation:

\[
134 = 7.9932a
\]

Now, solve for \( a \):

\[
a = \frac{134}{7.9932} \approx 16.77
\]

Rounding to the nearest whole number, we find:

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

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