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
If \( DE = 5 \), find \( AB \).
#### Explanation of the Diagram:
In the diagram, triangle \( ABC \) is provided with point \( D \) on side \( AC \) and point \( E \) on side \( AB \), such that segment \( DE \) is parallel to segment \( AB \). Given that segments \( AD \) and \( DC \) are congruent (or equal in length), and \( \overline{DE} \parallel \overline{AB} \), you need to find the length of segment \( AB \) if \( DE \) is equal to 5 units.
### Solution Approach:
To solve this problem, we can use properties of similar triangles. Since \( \overline{DE} \parallel \overline{AB} \), triangles \( ADE \) and \( ABC \) are similar by the Basic Proportionality Theorem (also known as Thales' Theorem).
Let's denote the length of \( AD \) (and \( DC \)) as \( x \) and the length of \( AB \) as \( y \).
From the similarity of triangles \( ADE \) and \( ABC \):
\[ \frac{AD}{AC} = \frac{DE}{AB} \]
Since \( AD = DC \),
\[ AC = AD + DC = 2AD = 2x \]
Substituting into the similarity ratio:
\[ \frac{AD}{2AD} = \frac{DE}{AB} \]
Simplifying:
\[ \frac{x}{2x} = \frac{5}{y} \]
\[ \frac{1}{2} = \frac{5}{y} \]
Cross-multiplying to solve for \( y \):
\[ y = 10 \]
Therefore, the length of \( AB \) is 10 units.
### Conclusion:
Using the properties of similar triangles, we determined that the length of \( AB \) is 10 units given the information in the diagram.
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### Educational Math Problem: Similar Triangles
#### Problem Statement:
In the diagram below, \( \overline{AD} \cong \overline{DC} \) and \( \overline{DE} \parallel \overline{AB} \).
If \( DE = 5 \), find \( AB \).
#### Explanation of the Diagram:
In the diagram, triangle \( ABC \) is provided with point \( D \) on side \( AC \) and point \( E \) on side \( AB \), such that segment \( DE \) is parallel to segment \( AB \). Given that segments \( AD \) and \( DC \) are congruent (or equal in length), and \( \overline{DE} \parallel \overline{AB} \), you need to find the length of segment \( AB \) if \( DE \) is equal to 5 units.
### Solution Approach:
To solve this problem, we can use properties of similar triangles. Since \( \overline{DE} \parallel \overline{AB} \), triangles \( ADE \) and \( ABC \) are similar by the Basic Proportionality Theorem (also known as Thales' Theorem).
Let's denote the length of \( AD \) (and \( DC \)) as \( x \) and the length of \( AB \) as \( y \).
From the similarity of triangles \( ADE \) and \( ABC \):
\[ \frac{AD}{AC} = \frac{DE}{AB} \]
Since \( AD = DC \),
\[ AC = AD + DC = 2AD = 2x \]
Substituting into the similarity ratio:
\[ \frac{AD}{2AD} = \frac{DE}{AB} \]
Simplifying:
\[ \frac{x}{2x} = \frac{5}{y} \]
\[ \frac{1}{2} = \frac{5}{y} \]
Cross-multiplying to solve for \( y \):
\[ y = 10 \]
Therefore, the length of \( AB \) is 10 units.
### Conclusion:
Using the properties of similar triangles, we determined that the length of \( AB \) is 10 units given the information in the diagram.
---
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