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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]\( ABC \)[/tex]?

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

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

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

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

Sagot :

GinnyAnswer
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]
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