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
Related questions
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc19db7c-4792-4fd2-b3d8-699ca1486fb3%2Feb13f157-20f7-42cf-a812-39e661dd67fa%2Fnizgm2_reoriented.jpeg&w=3840&q=75)
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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