Question

If triangle ABC =26 inches^2, a=5 inches, and angle C=60 degrees, then what is the measure of b to the nearest whole inch?

181 inches
19 inches
12 inches
16 inches

Answer 1

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

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

We are given:
- Area of triangle \( ABC = 26 \) square inches
- Side \( a = 5 \) inches
- Angle \( C = 60^\circ \)

Substituting these values into the area formula, we have:

\[
26 = \frac{1}{2} \cdot 5 \cdot b \cdot \sin(60^\circ)
\]

We know that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \).

Now substituting \( \sin(60^\circ) \):

\[
26 = \frac{1}{2} \cdot 5 \cdot b \cdot \frac{\sqrt{3}}{2}
\]

Simplifying further:

\[
26 = \frac{5b\sqrt{3}}{4}
\]

To isolate \( b \), we first multiply both sides by 4:

\[
104 = 5b\sqrt{3}
\]

Next, divide both sides by \( 5\sqrt{3} \):

\[
b = \frac{104}{5\sqrt{3}}
\]

Calculating the value of \( b \):

1. First, compute \( 5\sqrt{3} \):
- \( \sqrt{3} \approx 1.732 \)
- \( 5\sqrt{3} \approx 5 \times 1.732 = 8.66 \)

2. Now substitute this value into the equation for \( b \):

\[
b \approx \frac{104}{8.66} \approx 12.0
\]

Finally, rounding \( b \) to the nearest whole inch gives:

\[
\boxed{12}
\]

Thus, the measure of side \( b \) is approximately \( 12 \) inches.