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