Question

find the area of this triangle.
9 cm
110°
15 cm
? cm²
round your answer to the nearest hundredth.

Explanation:

Step1: Recall the formula for the area of a triangle with two sides and included angle

The formula for the area \( A \) of a triangle when two sides \( a \) and \( b \) and the included angle \( C \) are known is \( A=\frac{1}{2}ab\sin C \).
Here, \( a = 15\space\text{cm} \), \( b = 9\space\text{cm} \), and \( C = 110^{\circ} \).

Step2: Substitute the values into the formula

First, calculate \( \sin(110^{\circ}) \). We know that \( \sin(110^{\circ})=\sin(180^{\circ} - 70^{\circ})=\sin(70^{\circ})\approx0.9397 \).
Now substitute \( a = 15 \), \( b = 9 \), and \( \sin C\approx0.9397 \) into the formula:
\( A=\frac{1}{2}\times15\times9\times\sin(110^{\circ}) \)
\( A=\frac{1}{2}\times15\times9\times0.9397 \)
First, calculate \( 15\times9 = 135 \). Then, \( \frac{1}{2}\times135 = 67.5 \).
Now, \( 67.5\times0.9397\approx67.5\times0.9397 \)
\( 67.5\times0.9397 = 67.5\times(0.9 + 0.0397)=67.5\times0.9+67.5\times0.0397 = 60.75+2.68975 = 63.43975 \)

Step3: Round to the nearest hundredth

Rounding \( 63.43975 \) to the nearest hundredth gives \( 63.44 \).

Answer:

\( 63.44 \space\text{cm}^2 \)