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]
