Question

A triangle with three sides labeled as 25 centimeters, 24 centimeters, and 19 centimeters is shown. The angle opposite the side labeled 24 centimeters is labeled x degrees.
Question
Use the image to answer the question.
Solve for x in the triangle.
(1 point)
Responses

25.6
25.6

27.8
27.8

45.6
45.6

64.4

Answer 1

To solve for angle \( x \) in the triangle using the given side lengths, we can apply the Law of Cosines. The Law of Cosines states:

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

where \( c \) is the side opposite to angle \( C \), and \( a \) and \( b \) are the other two sides.

In your case:
- Let \( a = 25 \) cm (one side),
- Let \( b = 19 \) cm (the second side),
- Let \( c = 24 \) cm (the side opposite \( x \)).

Using the Law of Cosines to solve for \( \cos(x) \):

\[
24^2 = 25^2 + 19^2 - 2 \cdot 25 \cdot 19 \cdot \cos(x)
\]

Calculating \( 24^2 \), \( 25^2 \), and \( 19^2 \):

\[
576 = 625 + 361 - 950 \cdot \cos(x)
\]

Now, simplifying the right side:

\[
576 = 986 - 950 \cdot \cos(x)
\]

Rearranging to isolate \( \cos(x) \):

\[
950 \cdot \cos(x) = 986 - 576
\]
\[
950 \cdot \cos(x) = 410
\]
\[
\cos(x) = \frac{410}{950}
\]
\[
\cos(x) = 0.4316
\]

Now, we can find \( x \):

\[
x = \cos^{-1}(0.4316)
\]

Calculating \( x \):

\[
x \approx 64.4 \text{ degrees}
\]

Thus, the measure of angle \( x \) in the triangle is approximately \( 64.4 \) degrees.

The correct response is:

**64.4**