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Triangle ABC has the dimensions shown. A. D B 3.x-3 5x+1 E C

Triangle ABC has the dimensions shown. A. D B 3.x-3 5x+1 E C

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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### Triangle Proportions and Length Calculation #### Problem Description For the given triangle \( \triangle ABC \), certain dimensions are provided: - **Segment \( DE \)**, which is parallel to side \( AC \), has a length of \( 3x - 3 \) units. - **Base \( AC \)** has a length of \( 5x + 1 \) units. Given the proportional relationship between \( DE \) and \( AC \), the task is to determine the length of \( DE \). #### Diagram Explanation The diagram showcases a triangle \( \triangle ABC \): - Point \( D \) lies on segment \( AB \). - Point \( E \) lies on segment \( BC \). - Line segment \( DE \) is parallel to \( AC \), indicating the segments form similar triangles by the Basic Proportionality Theorem (Thales' theorem). #### Calculation To find the length of \( DE \), consider \( DE \) in relation to the entire base \( AC \). We use the provided expressions: 1. Base \( AC = 5x + 1 \) 2. \( DE = 3x - 3 \) #### Given Choices The possible lengths for \( DE \) are: - 7 units - 11 units - 18 units - 36 units #### Solution We are tasked to find the value \( x \) such that \( DE = 3x - 3 \). 1. Calculate \( x \) based on the dimensions provided. 2. Substitute back into the expression for \( DE = 3x - 3 \) to find the actual length. #### Answer This problem requires solving for \( x \) and then finding the correct length given the choices. **Options:** - 7 units - 11 units - 18 units - 36 units Compare your computed \( DE \) with the options to find the correct length. This exercise aids in understanding geometrical properties and proportionality within triangles, specifically the application of Thales' theorem in parallel line segments within triangles. ### Original Problem Statement - **What is the length of DE?** 1. ⃝ 7 units 2. ⃝ 11 units 3. ⃝ 18 units 4. ⃝ 36 units
Transcribed Image Text:### Triangle Proportions and Length Calculation #### Problem Description For the given triangle \( \triangle ABC \), certain dimensions are provided: - **Segment \( DE \)**, which is parallel to side \( AC \), has a length of \( 3x - 3 \) units. - **Base \( AC \)** has a length of \( 5x + 1 \) units. Given the proportional relationship between \( DE \) and \( AC \), the task is to determine the length of \( DE \). #### Diagram Explanation The diagram showcases a triangle \( \triangle ABC \): - Point \( D \) lies on segment \( AB \). - Point \( E \) lies on segment \( BC \). - Line segment \( DE \) is parallel to \( AC \), indicating the segments form similar triangles by the Basic Proportionality Theorem (Thales' theorem). #### Calculation To find the length of \( DE \), consider \( DE \) in relation to the entire base \( AC \). We use the provided expressions: 1. Base \( AC = 5x + 1 \) 2. \( DE = 3x - 3 \) #### Given Choices The possible lengths for \( DE \) are: - 7 units - 11 units - 18 units - 36 units #### Solution We are tasked to find the value \( x \) such that \( DE = 3x - 3 \). 1. Calculate \( x \) based on the dimensions provided. 2. Substitute back into the expression for \( DE = 3x - 3 \) to find the actual length. #### Answer This problem requires solving for \( x \) and then finding the correct length given the choices. **Options:** - 7 units - 11 units - 18 units - 36 units Compare your computed \( DE \) with the options to find the correct length. This exercise aids in understanding geometrical properties and proportionality within triangles, specifically the application of Thales' theorem in parallel line segments within triangles. ### Original Problem Statement - **What is the length of DE?** 1. ⃝ 7 units 2. ⃝ 11 units 3. ⃝ 18 units 4. ⃝ 36 units
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