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What is the range of possible sizes for side x?

What is the range of possible sizes for side x?

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Question

What is the range of possible sizes for side x?

### Understanding the Triangle Side Lengths #### Problem Statement: What is the range of possible sizes for side \( x \)? #### Diagram Explanation: The image depicts a triangle with three sides labeled as follows: - One side measures \( 4.1 \) units. - Another side measures \( 1.3 \) units. - The third side is labeled as \( x \) and its length is unknown. #### Explanation: To determine the range of possible sizes for side \( x \), we need to consider the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given the side lengths \( 4.1 \), \( 1.3 \), and \( x \), we can set up the following inequalities: 1. \( 4.1 + 1.3 > x \) 2. \( 4.1 + x > 1.3 \) 3. \( 1.3 + x > 4.1 \) Let's simplify these inequalities: 1. \( 5.4 > x \) or \( x < 5.4 \) 2. \( x > 1.3 - 4.1 \) or \( x > -2.8 \) (which is always true since side lengths are positive) 3. \( x > 4.1 - 1.3 \) or \( x > 2.8 \) Combining these results, we get the range: \[ 2.8 < x < 5.4 \] Thus, the range of possible sizes for side \( x \) is: \[ 2.8 \text{ units} < x < 5.4 \text{ units} \] --- This content aims to help students understand how to apply the triangle inequality theorem to determine the range of possible sizes for the sides of a triangle.
Transcribed Image Text:### Understanding the Triangle Side Lengths #### Problem Statement: What is the range of possible sizes for side \( x \)? #### Diagram Explanation: The image depicts a triangle with three sides labeled as follows: - One side measures \( 4.1 \) units. - Another side measures \( 1.3 \) units. - The third side is labeled as \( x \) and its length is unknown. #### Explanation: To determine the range of possible sizes for side \( x \), we need to consider the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given the side lengths \( 4.1 \), \( 1.3 \), and \( x \), we can set up the following inequalities: 1. \( 4.1 + 1.3 > x \) 2. \( 4.1 + x > 1.3 \) 3. \( 1.3 + x > 4.1 \) Let's simplify these inequalities: 1. \( 5.4 > x \) or \( x < 5.4 \) 2. \( x > 1.3 - 4.1 \) or \( x > -2.8 \) (which is always true since side lengths are positive) 3. \( x > 4.1 - 1.3 \) or \( x > 2.8 \) Combining these results, we get the range: \[ 2.8 < x < 5.4 \] Thus, the range of possible sizes for side \( x \) is: \[ 2.8 \text{ units} < x < 5.4 \text{ units} \] --- This content aims to help students understand how to apply the triangle inequality theorem to determine the range of possible sizes for the sides of a triangle.
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