A right triangle has side lengths [tex]\( AC = 7 \)[/tex] inches, [tex]\( BC = 24 \)[/tex] inches, and [tex]\( AB = 25 \)[/tex] inches. What are the measures of the angles in triangle [tex]\( \triangle ABC \)[/tex]?

A. [tex]\( m \angle A = 46.2^{\circ}, m \angle B = 43.8^{\circ}, m \angle C = 90^{\circ} \)[/tex]

B. [tex]\( m \angle A = 73.0^{\circ}, m \angle B = 17.0^{\circ}, m \angle C = 90^{\circ} \)[/tex]

C. [tex]\( m \angle A \approx 73.7^{\circ}, m \angle B = 16.3^{\circ}, m \angle C = 90^{\circ} \)[/tex]

D. [tex]\( m \angle A = 74.4^{\circ}, m \angle B = 15.6^{\circ}, m \angle C = 90^{\circ} \)[/tex]



Answer :

Given a right triangle with side lengths [tex]\(AC = 7\)[/tex] inches, [tex]\(BC = 24\)[/tex] inches, and [tex]\(AB = 25\)[/tex] inches, we need to find the measures of the angles in triangle [tex]\(ABC\)[/tex].

First, let's note the relationship between the sides and the angles. In any right triangle:
- The hypotenuse is the longest side, which is [tex]\(AB = 25\)[/tex] inches in this case.
- [tex]\(AC\)[/tex] and [tex]\(BC\)[/tex] are the legs of the triangle.

Since angle [tex]\(C\)[/tex] is the right angle ([tex]\(90^\circ\)[/tex]), we need to calculate the measures of angles [tex]\(A\)[/tex] and [tex]\(B\)[/tex].

Using the cosine rule, we can find these angles step-by-step:

### Calculating [tex]\(\angle A\)[/tex]:

The cosine of angle [tex]\(A\)[/tex], adjacent to side [tex]\(BC\)[/tex] and opposite side [tex]\(AC\)[/tex], is given by the formula:
[tex]\[ \cos(A) = \frac{BC^2 + AB^2 - AC^2}{2 \cdot BC \cdot AB} \][/tex]

Substituting the given values:
[tex]\[ \cos(A) = \frac{24^2 + 25^2 - 7^2}{2 \cdot 24 \cdot 25} \][/tex]

[tex]\[ \cos(A) = \frac{576 + 625 - 49}{1200} \][/tex]

[tex]\[ \cos(A) = \frac{1152}{1200} \][/tex]

[tex]\[ \cos(A) = 0.96 \][/tex]

Therefore, [tex]\(\angle A\)[/tex] can be calculated using the arccosine function:
[tex]\[ \angle A \approx \cos^{-1}(0.96) \approx 16.3^\circ \][/tex]

### Calculating [tex]\(\angle B\)[/tex]:

Similarly, the cosine of angle [tex]\(B\)[/tex], adjacent to side [tex]\(AC\)[/tex] and opposite side [tex]\(BC\)[/tex], is given by the formula:
[tex]\[ \cos(B) = \frac{AC^2 + AB^2 - BC^2}{2 \cdot AC \cdot AB} \][/tex]

Substituting the given values:
[tex]\[ \cos(B) = \frac{7^2 + 25^2 - 24^2}{2 \cdot 7 \cdot 25} \][/tex]

[tex]\[ \cos(B) = \frac{49 + 625 - 576}{350} \][/tex]

[tex]\[ \cos(B) = \frac{98}{350} \][/tex]

[tex]\[ \cos(B) = 0.28 \][/tex]

Therefore, [tex]\(\angle B\)[/tex] can be calculated using the arccosine function:
[tex]\[ \angle B \approx \cos^{-1}(0.28) \approx 73.7^\circ \][/tex]

### Conclusion:

We have calculated the angles in triangle [tex]\(ABC\)[/tex] as approximately:
[tex]\[ \angle A \approx 16.3^\circ \][/tex]
[tex]\[ \angle B \approx 73.7^\circ \][/tex]
[tex]\[ \angle C = 90^\circ \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{m \angle A \approx 16.3^\circ, m \angle B \approx 73.7^\circ, m \angle C = 90^\circ} \][/tex]