Question

If the area of triangle ABC =27cm^2, m angle C=40 degrees, and side b=14 cm, then what is the measure of side a? Round your answer to the nearest whole number.

6cm
5cm
3cm
2cm

Answer 1

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

\[
\text{Area} = \frac{1}{2} \times b \times a \times \sin(C)
\]

Where:
- Area = \( 27 \text{ cm}^2 \)
- \( b = 14 \text{ cm} \)
- \( C = 40^\circ \)

First, we can substitute the known values into the formula:

\[
27 = \frac{1}{2} \times 14 \times a \times \sin(40^\circ)
\]

Now, let's calculate \( \sin(40^\circ) \). Using a scientific calculator, we find:

\[
\sin(40^\circ) \approx 0.6428
\]

Substituting this back into the formula gives us:

\[
27 = \frac{1}{2} \times 14 \times a \times 0.6428
\]

Now, solve for \( a \):

1. Multiply both sides by 2:

\[
54 = 14 \times a \times 0.6428
\]

2. Dividing both sides by \( 14 \times 0.6428 \):

\[
a = \frac{54}{14 \times 0.6428}
\]

Calculating the denominator:

\[
14 \times 0.6428 \approx 8.9992
\]

Now substituting this value in:

\[
a = \frac{54}{8.9992} \approx 6.0008
\]

Rounding to the nearest whole number gives:

\[
\boxed{6}
\]

So the measure of side \( a \) is approximately \( 6 \text{ cm} \).