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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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This diagram illustrates a geometric problem involving a triangle inside another triangle and the concept of similar triangles. 1. **Diagram Description**: - The outer triangle is \( \triangle BCD \) with vertices labeled \( B, C, D \). - The point \( A \) is on side \( CD \) and divides \( CD \) into two segments: \( CA \) and \( AD \). - A line segment is drawn from point \( B \) to point \( A \), creating two smaller triangles, \( \triangle ABC \) and \( \triangle ABD \). - There are two ticks on segments \( AD \) and \( CA \), indicating that these two segments are equal in length. 2. **Given Information**: - A variable \( x \) is placed near segment \( AD \), suggesting that the length of \( AD \) is represented by \( x \). 3. **Key Characteristics**: - The two triangles \( \triangle ABD \) and \( \triangle ABC \) share the common vertex \( B \) and the side \( AB \). - The marks on \( AD \) and \( AC \) indicate that these segments are congruent. ### Explanation: The diagram is used typically for problems requiring an understanding of properties such as similarity and congruence. The equal segments signify that \( A \) is the midpoint of \( CD \), thus making \( \triangle ABD \) and \( \triangle ABC \) potentially similar, depending on the context of angles or other further relations that might be provided. Such geometric problems often serve as exercises to test the understanding of triangle properties, midpoints, and the Side-Splitter Theorem, among others.
Transcribed Image Text:This diagram illustrates a geometric problem involving a triangle inside another triangle and the concept of similar triangles. 1. **Diagram Description**: - The outer triangle is \( \triangle BCD \) with vertices labeled \( B, C, D \). - The point \( A \) is on side \( CD \) and divides \( CD \) into two segments: \( CA \) and \( AD \). - A line segment is drawn from point \( B \) to point \( A \), creating two smaller triangles, \( \triangle ABC \) and \( \triangle ABD \). - There are two ticks on segments \( AD \) and \( CA \), indicating that these two segments are equal in length. 2. **Given Information**: - A variable \( x \) is placed near segment \( AD \), suggesting that the length of \( AD \) is represented by \( x \). 3. **Key Characteristics**: - The two triangles \( \triangle ABD \) and \( \triangle ABC \) share the common vertex \( B \) and the side \( AB \). - The marks on \( AD \) and \( AC \) indicate that these segments are congruent. ### Explanation: The diagram is used typically for problems requiring an understanding of properties such as similarity and congruence. The equal segments signify that \( A \) is the midpoint of \( CD \), thus making \( \triangle ABD \) and \( \triangle ABC \) potentially similar, depending on the context of angles or other further relations that might be provided. Such geometric problems often serve as exercises to test the understanding of triangle properties, midpoints, and the Side-Splitter Theorem, among others.
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