Question

A triangle is shown with three sides labeled as 7 feet, 11 feet, and 8 feet. The angle across from the 11 foot side is labeled x.
Question
Use the image to answer the question.
Solve the triangle for x.
(1 point)
Responses

39.4°
39.4°

46.5°
46.5°

94.1°
94.1°

105.9°
105.9°
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Answer 1

To find the angle \( x \) across from the side measuring \( 11 \) feet in a triangle with sides of \( 7 \) feet, \( 8 \) feet, and \( 11 \) feet, we can use the Law of Cosines.

The Law of Cosines states:

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

where:
- \( c \) is the length of the side opposite the angle \( X \).
- \( a \) and \( b \) are the lengths of the other two sides.

In this case:
- \( c = 11 \) feet,
- \( a = 7 \) feet,
- \( b = 8 \) feet.

We will solve for \( x \):

\[
11^2 = 7^2 + 8^2 - 2 \cdot 7 \cdot 8 \cdot \cos(x)
\]

Calculating the squares:

\[
121 = 49 + 64 - 112 \cdot \cos(x)
\]

Adding \( 49 + 64 \):

\[
121 = 113 - 112 \cdot \cos(x)
\]

Now, we isolate the cosine term:

\[
121 - 113 = -112 \cdot \cos(x)
\]

\[
8 = -112 \cdot \cos(x)
\]

Dividing both sides by \(-112\):

\[
\cos(x) = -\frac{8}{112} = -\frac{1}{14}
\]

Now, we can find \( x \) using the inverse cosine function:

\[
x = \cos^{-1}\left(-\frac{1}{14}\right)
\]

Calculating that gives approximately:

\[
x \approx 105.9^\circ
\]

Therefore, the answer is \( 105.9^\circ \). Thus, the final response is:

**105.9°**