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Find the length of side xx in simplest radical form with a rational denominator

Find the length of side xx in simplest radical form with a rational denominator

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Find the length of side xx in simplest radical form with a rational denominator.

**Problem Statement:** Find the length of side \( x \) in simplest radical form with a rational denominator. **Diagram Description:** The image shows a right triangle with the following attributes: - One angle is \( 30^\circ \). - Another angle is \( 60^\circ \). - The right angle is marked with a small square. - The side opposite the \( 30^\circ \) angle (hypotenuse) is labeled as \( x \). - The side opposite the \( 60^\circ \) angle is labeled as \( 3 \). **Solution Explanation:** 1. **Understanding the 30-60-90 Triangle:** A 30-60-90 triangle has sides in the ratio \( 1 : \sqrt{3} : 2 \). - The side opposite the \( 30^\circ \) angle (shortest side) is \( a \). - The side opposite the \( 60^\circ \) angle (longest leg) is \( a\sqrt{3} \). - The hypotenuse is \( 2a \). 2. **Given Information:** - The side opposite the \( 60^\circ \) angle \( = 3 \). 3. **Finding \( a \):** - According to the ratio, \( a\sqrt{3} = 3 \). - Solving for \( a \): \( a = \frac{3}{\sqrt{3}} \). - Rationalize the denominator: \( a = \frac{3}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \sqrt{3} \). 4. **Finding \( x \) (the hypotenuse):** - The hypotenuse \( x = 2a = 2\sqrt{3} \). Thus, the length of side \( x \) is \( 2\sqrt{3} \).
Transcribed Image Text:**Problem Statement:** Find the length of side \( x \) in simplest radical form with a rational denominator. **Diagram Description:** The image shows a right triangle with the following attributes: - One angle is \( 30^\circ \). - Another angle is \( 60^\circ \). - The right angle is marked with a small square. - The side opposite the \( 30^\circ \) angle (hypotenuse) is labeled as \( x \). - The side opposite the \( 60^\circ \) angle is labeled as \( 3 \). **Solution Explanation:** 1. **Understanding the 30-60-90 Triangle:** A 30-60-90 triangle has sides in the ratio \( 1 : \sqrt{3} : 2 \). - The side opposite the \( 30^\circ \) angle (shortest side) is \( a \). - The side opposite the \( 60^\circ \) angle (longest leg) is \( a\sqrt{3} \). - The hypotenuse is \( 2a \). 2. **Given Information:** - The side opposite the \( 60^\circ \) angle \( = 3 \). 3. **Finding \( a \):** - According to the ratio, \( a\sqrt{3} = 3 \). - Solving for \( a \): \( a = \frac{3}{\sqrt{3}} \). - Rationalize the denominator: \( a = \frac{3}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \sqrt{3} \). 4. **Finding \( x \) (the hypotenuse):** - The hypotenuse \( x = 2a = 2\sqrt{3} \). Thus, the length of side \( x \) is \( 2\sqrt{3} \).
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