If the area of △ABC=134 △ 𝐴 𝐵 𝐶 = 134 square feet, m∠C = 41° 𝑚 ∠ 𝐶 = 41 ° , and b=24 𝑏 = 24 feet, then what is the length of side a 𝑎 ? Round your answer to the nearest whole number. (1 point) Responses 15 feet 15 feet 17 feet 17 feet 16 feet 16 feet 9 feet

Answer 1

To find the length of side \( a \) in triangle \( ABC \), we can use the formula for the area of a triangle given two sides and the included angle:

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

Where:

• \( A \) is the area of the triangle,
• \( a \) and \( b \) are the lengths of two sides,
• \( C \) is the included angle.

From the given information:

• Area \( A = 134 \) square feet,
• Side \( b = 24 \) feet,
• Angle \( C = 41^\circ \).

We can rearrange the formula to solve for \( a \):

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

First, compute \( \sin(41^\circ) \):

\[ \sin(41^\circ) \approx 0.6561 \quad(\text{using a calculator}) \]

Now substitute \( \sin(41^\circ) \) into the equation:

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

This simplifies to:

\[ 134 = 12a \cdot 0.6561 \] \[ 134 = 7.8732a \]

Now, solve for \( a \):

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

Rounding to the nearest whole number gives:

\[ \boxed{17} \text{ feet} \]