Question

If in oblique ΔABC side a=13 m, side b=6 m, and side c=9 m, then use the Law of Cosines to find the measure of ∠A .(1 point)
Responses

63 degrees
63 degrees

50 degrees
50 degrees

119 degrees
119 degrees

15 degrees
15 degrees

Answer 1

To find the measure of angle \(A\) in triangle \(ABC\) using the Law of Cosines, we can use the following formula:

\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]

Given:
- \(a = 13\) m
- \(b = 6\) m
- \(c = 9\) m

First, we will calculate \(b^2 + c^2 - a^2\):

\[
b^2 = 6^2 = 36
\]
\[
c^2 = 9^2 = 81
\]
\[
a^2 = 13^2 = 169
\]

Now substitute into the equation:

\[
b^2 + c^2 - a^2 = 36 + 81 - 169 = 117 - 169 = -52
\]

Now calculate \(2bc\):

\[
2bc = 2 \times 6 \times 9 = 108
\]

Now plug these values into the Law of Cosines formula:

\[
\cos(A) = \frac{-52}{108}
\]

Calculating this gives:

\[
\cos(A) = -\frac{52}{108} \approx -0.4815
\]

Now, to find \(A\), take the arccosine:

\[
A \approx \cos^{-1}(-0.4815)
\]

Using a calculator:

\[
A \approx 118.93^\circ
\]

Rounding to the nearest degree, we find:

\[
A \approx 119^\circ
\]

Thus, the measure of \(\angle A\) is \(119\) degrees. The correct response is:

**119 degrees**