In the diagram, it is given that \( BD = a \), \( BC = b \), and \( BA = 1 \). Line \( AC \) is parallel to line \( DE \).**Description of the Diagram:**- The diagram presents a geometric figure with points labeled \( B \), \( A \), \( C \), \( D \), and \( E \).- The lines \( BA \), \( AC \), \( BD \), and \( DE \) are depicted, forming a combination of triangles and quadrilateral shapes.- Line \( BD \) is marked with length \( a \), line \( BC \), parallel to line \( BE \), is marked with length \( b \), and line \( BA \) is marked with length \( 1 \).- Line \( AC \) is parallel to line \( DE \), suggesting that \( \triangle BAC \sim \triangle ADE \) by the basic proportionality theorem or the concept of similarity.**Goal:**Prove that the length of \( BE = ab \).This setup calls for the use of geometric properties and similarity of triangles to derive and prove the relationship stated. Such problems typically involve manipulating the relationships between similar triangles or parallel lines to reach the desired equation.

Answered: In the figure, suppose BD = a, BC = b,…

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In the figure, suppose BD = a, BC = b, and BA = 1. Assume AC is parallel to DE B Prove: BE = ab. A C D

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

In the diagram, it is given that \( BD = a \), \( BC = b \), and \( BA = 1 \). Line \( AC \) is parallel to line \( DE \).

**Description of the Diagram:**

- The diagram presents a geometric figure with points labeled \( B \), \( A \), \( C \), \( D \), and \( E \).
- The lines \( BA \), \( AC \), \( BD \), and \( DE \) are depicted, forming a combination of triangles and quadrilateral shapes.
- Line \( BD \) is marked with length \( a \), line \( BC \), parallel to line \( BE \), is marked with length \( b \), and line \( BA \) is marked with length \( 1 \).
- Line \( AC \) is parallel to line \( DE \), suggesting that \( \triangle BAC \sim \triangle ADE \) by the basic proportionality theorem or the concept of similarity.

**Goal:**

Prove that the length of \( BE = ab \).

This setup calls for the use of geometric properties and similarity of triangles to derive and prove the relationship stated. Such problems typically involve manipulating the relationships between similar triangles or parallel lines to reach the desired equation.

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