Question

Find the value of x in the triangle below:
(5 points)
sides are 11 5 15
Responses

130°
130 degrees

132°
132 degrees

136°
136 degrees

139°
139 degrees

Answer 1

To find the value of \( x \) in the triangle with sides 11, 5, and 15, we can use the Law of Cosines. According to this theorem, for a triangle with sides \( a \), \( b \), and \( c \) opposite to angles \( A \), \( B \), and \( C \) respectively:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Let's assign the sides as follows:
- \( a = 11 \)
- \( b = 5 \)
- \( c = 15 \)

We want to find angle \( C \) which is opposite side \( c = 15 \).

Applying the Law of Cosines, we have:

\[
15^2 = 11^2 + 5^2 - 2 \cdot 11 \cdot 5 \cdot \cos(C)
\]

Calculating the squares:

\[
225 = 121 + 25 - 110 \cdot \cos(C)
\]

Combining the constants:

\[
225 = 146 - 110 \cdot \cos(C)
\]

Rearranging the equation:

\[
110 \cdot \cos(C) = 146 - 225
\]
\[
110 \cdot \cos(C) = -79
\]
\[
\cos(C) = \frac{-79}{110}
\]

Now, we calculate the angle \( C \):

\[
C = \cos^{-1}\left(\frac{-79}{110}\right)
\]

Using a calculator, we find:

\[
C \approx 130.4°
\]

This suggests that \( C \) fits with one of the response options. The closest available response is \( 130° \).

Therefore, the value of \( x \) is:

**130°**