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

119 degrees
119 degrees

15 degrees
15 degrees

63 degrees
63 degrees

50 degrees

Answer 1

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

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

Where:
- \( a \) is the length of side \( a \) (opposite angle \( A \)),
- \( b \) is the length of side \( b \),
- \( c \) is the length of side \( c \).

Given:
- \( a = 13 \, \text{m} \)
- \( b = 6 \, \text{m} \)
- \( c = 9 \, \text{m} \)

Substituting the values into the formula:

\[
\cos A = \frac{6^2 + 9^2 - 13^2}{2 \cdot 6 \cdot 9}
\]

Calculating each component:

\[
6^2 = 36,
\]
\[
9^2 = 81,
\]
\[
13^2 = 169.
\]

Now substitute these values into the equation:

\[
\cos A = \frac{36 + 81 - 169}{2 \cdot 6 \cdot 9}
\]
\[
= \frac{117 - 169}{108}
\]
\[
= \frac{-52}{108}
\]
\[
= \frac{-26}{54}
\]
\[
= \frac{-13}{27}.
\]

Now we take the inverse cosine to find angle \( A \):

\[
A = \cos^{-1}\left(-\frac{13}{27}\right).
\]

Calculating \( A \):

Using a calculator:
\[
A \approx 119 \, \text{degrees}.
\]

So the measure of angle \( A \) is approximately \( 119 \, \text{degrees} \).

**Final answer: 119 degrees.**