Question

USE The Law of Sines

TRIANGLE:
AC: 15 in
BC: 10 in
B: 85 degree
C:40 degree
Use the Law of Sines to find the value of side length c. Round to the nearest tenth of an inch

a. 0.2 inches
b. 6.5 inches
c. 9.7 inches
d. 15.5 inches

Answer 1

To find the length of side \( c \) in triangle ABC using the Law of Sines, we start by identifying the sides and angles we have. We have:

- Side \( AC = b = 15 \) inches
- Side \( BC = a = 10 \) inches
- Angle \( B = 85^\circ \)
- Angle \( C = 40^\circ \)

First, we need to find angle \( A \). We can use the fact that the sum of the angles in a triangle is \( 180^\circ \):

\[
A = 180^\circ - B - C = 180^\circ - 85^\circ - 40^\circ = 55^\circ
\]

Now we can use the Law of Sines, which states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

We will use the ratios to find \( c \):

\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]

First, we calculate \( \sin A \) and \( \sin C \):

\[
\sin A = \sin(55^\circ), \quad \sin C = \sin(40^\circ)
\]

Now, we can set up the equation to find \( c \):

\[
\frac{10}{\sin(55^\circ)} = \frac{c}{\sin(40^\circ)}
\]

Now rearranging gives:

\[
c = 10 \cdot \frac{\sin(40^\circ)}{\sin(55^\circ)}
\]

Now we need to calculate the sine values and substitute them into the equation:

1. \( \sin(55^\circ) \approx 0.8192 \)
2. \( \sin(40^\circ) \approx 0.6428 \)

Substituting these values in:

\[
c = 10 \cdot \frac{0.6428}{0.8192} \approx 10 \cdot 0.7830 \approx 7.83 \text{ inches}
\]

This rounds to:

\[
c \approx 7.8 \text{ inches}
\]

Since the closest option is \( 9.7 \) inches, there may be a mistake in calculation selection or provided data. However, according to the calculations based on the values (rounded), none of the provided options are accurate. The calculated value is approximately **7.8 inches**.

Please check the original triangle parameters and re-confirm if there are any adjustments needed in the angle or side lengths.