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 . \[ ?A. \[ ? A . \[ M \angle A = 46.2^{\circ}, M \angle

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

Introduction

In this article, we will explore the problem of finding the measures of the angles in a right triangle when the side lengths are given. We will use the Pythagorean theorem and trigonometric ratios to solve for the angles.

The Problem

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 $ABC$?

Using the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, we have:

$$AB^2 = AC^2 + BC^2$$

Plugging in the given values, we get:

$$25^2 = 7^2 + 24^2$$

Simplifying, we get:

$$625 = 49 + 576$$

This equation is true, so we can conclude that the given side lengths form a right triangle.

Finding the Measures of the Angles

To find the measures of the angles, we can use the trigonometric ratios. Let's start by finding the measure of angle $A$.

We know that $\sin A = \frac{BC}{AB} = \frac{24}{25}$. Using a calculator, we can find the measure of angle $A$:

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

Using a calculator, we get:

$$m \angle A = 46.2^{\circ}$$

Q&A

Q: What is the measure of angle $B$?

A: To find the measure of angle $B$, we can use the fact that the sum of the measures of the angles in a triangle is $180^{\circ}$. We know that $m \angle A = 46.2^{\circ}$, so we can set up the equation:

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

Substituting the known value, we get:

$$46.2^{\circ} + m \angle B + 90^{\circ} = 180^{\circ}$$

Simplifying, we get:

$$m \angle B = 43.8^{\circ}$$

Q: What is the measure of angle $C$?

A: We know that $m \angle A = 46.2^{\circ}$ and $m \angle B = 43.8^{\circ}$. Using the fact that the sum of the measures of the angles in a triangle is $180^{\circ}$, we can find the measure of angle $C$:

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

Substituting the known values, we get:

$$46.2^{\circ} + 43.8^{\circ} + m \angle C = 180^{\circ}$$

Simplifying, we get:

$$m \angle C = 90^{\circ}$$

Q: What is the relationship between the side lengths and the angles in a right triangle?

A: In a right triangle, the side lengths are related to the angles through the trigonometric ratios. Specifically, the sine, cosine, and tangent of an angle are defined as:

$$\sin A = \frac{BC}{AB}$$

$$\cos A = \frac{AC}{AB}$$

$$\tan A = \frac{BC}{AC}$$

These ratios can be used to find the measures of the angles in a right triangle.

Q: How can we use the Pythagorean theorem to find the side lengths of a right triangle?

A: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:

$$AB^2 = AC^2 + BC^2$$

This equation can be used to find the side lengths of a right triangle, given the lengths of the other two sides.

Q: What is the significance of the Pythagorean theorem in mathematics?

A: The Pythagorean theorem is a fundamental concept in mathematics, with far-reaching implications in geometry, trigonometry, and other areas of mathematics. It is used to find the side lengths of right triangles, and is a key tool in solving problems involving right triangles.

Q: How can we use the trigonometric ratios to find the measures of the angles in a right triangle?

A: The trigonometric ratios can be used to find the measures of the angles in a right triangle. Specifically, the sine, cosine, and tangent of an angle are defined as:

$$\sin A = \frac{BC}{AB}$$

$$\cos A = \frac{AC}{AB}$$

$$\tan A = \frac{BC}{AC}$$

These ratios can be used to find the measures of the angles in a right triangle, given the side lengths of the triangle.