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A right triangle has side lengths AC = 7 inches, BC = 24 inches, and AB = 25 inches.

What are the measures of the angles in triangle ABC?

A. [tex]\( m \angle A \approx 46.2^\circ, \, m \angle B \approx 43.8^\circ, \, m \angle C = 90^\circ \)[/tex]
B. [tex]\( m \angle A \approx 73.0^\circ, \, m \angle B \approx 17.0^\circ, \, m \angle C = 90^\circ \)[/tex]
C. [tex]\( m \angle A = 73.7^\circ, \, m \angle B = 16.3^\circ, \, m \angle C = 90^\circ \)[/tex]
D. [tex]\( m \angle A = 74.4^\circ, \, m \angle B = 15.6^\circ, \, m \angle C = 90^\circ \)[/tex]



Answer :

GinnyAnswer
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]

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