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. $m \angle A \approx 46.2^{\circ}, \, M \angle B \approx

Feb 26, 2025 by ADMIN 203 views

A Right Triangle with Known Side Lengths: Finding the Measures of Angles

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Introduction

In this article, we will explore the concept of finding the measures of angles in a right triangle when the side lengths are known. We will use the given side lengths of triangle $ABC$ to find the measures of angles $A$ and $B$ . This problem is a classic example of applying trigonometric concepts to solve a real-world problem.

The Law of Cosines

The Law of Cosines is a fundamental concept in trigonometry that relates the side lengths of a triangle to the cosine of one of its angles. The Law of Cosines states that for any triangle with side lengths $a$ , $b$ , and $c$ , and angle $C$ opposite side $c$ , the following equation holds:

c^2 = a^2 + b^2 - 2ab \cos C

We can use this equation to find the measure of angle $C$ in triangle $ABC$ .

Applying the Law of Cosines to Triangle $ABC$

We are given the side lengths of triangle $ABC$ as $AC = 7$ inches, $BC = 24$ inches, and $AB = 25$ inches. We want to find the measures of angles $A$ and $B$ . To do this, we can use the Law of Cosines to find the measure of angle $C$ , which is the right angle in this triangle.

Using the Law of Cosines, we have:

AB^2 = AC^2 + BC^2 - 2(AC)(BC) \cos C

Plugging in the given side lengths, we get:

25^2 = 7^2 + 24^2 - 2(7)(24) \cos C

Simplifying the equation, we get:

625 = 49 + 576 - 336 \cos C

Combine like terms:

625 = 625 - 336 \cos C

Subtract 625 from both sides:

0 = -336 \cos C

Divide both sides by -336:

\cos C = 0

Since $\cos C = 0$ , we know that angle $C$ is a right angle, which means that $C = 90^{\circ}$ .

Finding the Measures of Angles $A$ and $B$

Now that we know the measure of angle $C$ , we can use the fact that the sum of the measures of the angles in a triangle is $180^{\circ}$ to find the measures of angles $A$ and $B$ .

We have:

m \angle A + m \angle B + m \angle C = 180^{\circ}

Since $m \angle C = 90^{\circ}$ , we can simplify the equation to:

m \angle A + m \angle B = 90^{\circ}

We can use the Law of Sines to find the measures of angles $A$ and $B$ . The Law of Sines states that for any triangle with side lengths $a$ , $b$ , and $c$ , and angles $A$ , $B$ , and $C$ , the following equation holds:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

We can use this equation to find the measures of angles $A$ and $B$ .

Applying the Law of Sines to Triangle $ABC$

Using the Law of Sines, we have:

\frac{AC}{\sin A} = \frac{BC}{\sin B}

Plugging in the given side lengths, we get:

\frac{7}{\sin A} = \frac{24}{\sin B}

We can rearrange this equation to solve for $\sin A$ :

\sin A = \frac{7 \sin B}{24}

We can also use the Law of Sines to find the measure of angle $B$ :

\frac{BC}{\sin B} = \frac{AB}{\sin C}

Plugging in the given side lengths, we get:

\frac{24}{\sin B} = \frac{25}{\sin 90^{\circ}}

Simplifying the equation, we get:

\sin B = \frac{24 \sin 90^{\circ}}{25}

Since $\sin 90^{\circ} = 1$ , we have:

\sin B = \frac{24}{25}

We can use this value to find the measure of angle $B$ :

m \angle B = \sin^{-1} \left(\frac{24}{25}\right)

Using a calculator, we get:

m \angle B \approx 76.0^{\circ}

Now that we have the measure of angle $B$ , we can use the fact that the sum of the measures of the angles in a triangle is $180^{\circ}$ to find the measure of angle $A$ :

m \angle A + m \angle B = 90^{\circ}

Substituting the value of $m \angle B$ , we get:

m \angle A + 76.0^{\circ} = 90^{\circ}