To determine the measures of the angles in the right triangle given the side lengths [tex]\( AC = 7 \)[/tex] inches, [tex]\( BC = 24 \)[/tex] inches, and [tex]\( AB = 25 \)[/tex] inches, follow these steps:
1. Identify the Hypotenuse and Legs:
- [tex]\( AB = 25 \)[/tex] inches (hypotenuse)
- [tex]\( AC = 7 \)[/tex] inches (opposite side to [tex]\( \angle B \)[/tex])
- [tex]\( BC = 24 \)[/tex] inches (adjacent side to [tex]\( \angle A \)[/tex])
2. Calculate the Angles Using Trigonometric Ratios:
- Angle [tex]\( A \)[/tex]:
[tex]\[
\sin \angle A = \frac{AC}{AB} = \frac{7}{25}
\][/tex]
Using the inverse sine function ([tex]\(\sin^{-1}\)[/tex]):
[tex]\[
\angle A = \sin^{-1}\left(\frac{7}{25}\right)
\][/tex]
- Angle [tex]\( B \)[/tex]:
[tex]\[
\cos \angle B = \frac{BC}{AB} = \frac{24}{25}
\][/tex]
Using the inverse cosine function ([tex]\(\cos^{-1}\)[/tex]):
[tex]\[
\angle B = \cos^{-1}\left(\frac{24}{25}\right)
\][/tex]
- Angle [tex]\( C \)[/tex]:
Since it's a right triangle, [tex]\( \angle C \)[/tex] is [tex]\( 90^\circ \)[/tex].
3. Determine Approximate Angle Values:
Using the provided trigonometric calculations:
[tex]\[
\angle A \approx 16.3^\circ
\][/tex]
[tex]\[
\angle B \approx 73.7^\circ
\][/tex]
[tex]\[
\angle C = 90^\circ
\][/tex]
4. Conclusion:
The measures of the angles in the triangle are:
- [tex]\( m \angle A = 16.3^\circ \)[/tex]
- [tex]\( m \angle B = 73.7^\circ \)[/tex]
- [tex]\( m \angle C = 90^\circ \)[/tex]
Thus, the correct answer from the given choices is:
[tex]\[ m \angle A = 16.3^\circ, m \angle B = 73.7^\circ, m \angle C = 90^\circ \][/tex]
