Alright, let's solve this problem step-by-step.
1. Identify the sides of the right triangle:
- [tex]\(AC = 7\)[/tex] inches
- [tex]\(BC = 24\)[/tex] inches
- [tex]\(AB = 25\)[/tex] inches
2. Identify the right angle:
- Since [tex]\(AB\)[/tex] is the hypotenuse (the longest side) in a right triangle, the right angle must be at [tex]\(C\)[/tex].
- Therefore, [tex]\(m \angle C = 90^\circ\)[/tex].
3. Calculate one of the angles using the cosine law for triangle [tex]\(ABC\)[/tex]:
Recall the law of cosines which states:
[tex]\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\][/tex]
In our triangle, let [tex]\(\angle A\)[/tex] be the angle opposite side [tex]\(BC = 24\)[/tex]:
[tex]\[
\cos(A) = \frac{BC^2 + AB^2 - AC^2}{2 \cdot BC \cdot AB}
\][/tex]
Substitute the known values [tex]\(BC = 24\)[/tex], [tex]\(AB = 25\)[/tex], and [tex]\(AC = 7\)[/tex]:
[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]
Now we find [tex]\(\angle A\)[/tex] by taking the arccosine of 0.96:
[tex]\[
\angle A \approx \arccos(0.96) \approx 16.3^\circ
\][/tex]
4. Use the fact that the sum of the angles in a triangle is [tex]\(180^\circ\)[/tex]:
[tex]\[
m \angle A + m \angle B + m \angle C = 180^\circ
\][/tex]
Substituting the known values:
[tex]\[
16.3^\circ + m \angle B + 90^\circ = 180^\circ
\][/tex]
Therefore,
[tex]\[
m \angle B = 180^\circ - 90^\circ - 16.3^\circ = 73.7^\circ
\][/tex]
So, the measures of the angles in triangle [tex]\(ABC\)[/tex] are:
- [tex]\(m \angle A \approx 16.3^\circ\)[/tex]
- [tex]\(m \angle B \approx 73.7^\circ\)[/tex]
- [tex]\(m \angle C = 90^\circ\)[/tex]
The correct choice from the given options is:
[tex]\[ m \angle A \approx 73.7^\circ, m \angle B \approx 16.3^\circ, m \angle C \approx 90^\circ \][/tex]
