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Given: G is the centroid of AAEC, EBL AC, AB= 6, AF = 5, EG = 5, and AE = CE. EB B

Given: G is the centroid of AAEC, EBL AC, AB= 6, AF = 5, EG = 5, and AE = CE. EB B

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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Given: G is the centroid of \(\Delta AEC\), \[ EB \perp AC \] \[ AB = 6 \] \[ AF = 5 \] \[ EG = 5 \frac{1}{3} \] \[ and \ AE = CE. \] EB = __________ Below this information is a geometric diagram of triangle \( \Delta AEC \) with the points labeled as follows: - Point \( A \) at the bottom left vertex. - Point \( B \) in the middle of the base \( AC \), forming a right angle with \( AC \). - Point \( C \) at the bottom right vertex. - Point \( E \) at the top vertex. - The centroid \( G \) of \(\Delta AEC\) is marked inside the triangle. - Line segments \( AE \) and \( CE \) are equal, implying the triangle is isosceles. - Point \( F \) is marked on \( AE \) with \( AF = 5 \). - Point \( D \) is marked on \( CE \). - \( EB \) is perpendicular to \( AC \) and line segment \( EB \) splits \( \Delta AEC \) into two right triangles. Analysis: - The centroid \( G \) divides each median into a 2:1 ratio, with the longer segment being between the vertex and the centroid. - \( EB \) is a perpendicular height from \( E \) to \( AC \). The goal of this problem is to find the length of \( EB \).
Transcribed Image Text:Given: G is the centroid of \(\Delta AEC\), \[ EB \perp AC \] \[ AB = 6 \] \[ AF = 5 \] \[ EG = 5 \frac{1}{3} \] \[ and \ AE = CE. \] EB = __________ Below this information is a geometric diagram of triangle \( \Delta AEC \) with the points labeled as follows: - Point \( A \) at the bottom left vertex. - Point \( B \) in the middle of the base \( AC \), forming a right angle with \( AC \). - Point \( C \) at the bottom right vertex. - Point \( E \) at the top vertex. - The centroid \( G \) of \(\Delta AEC\) is marked inside the triangle. - Line segments \( AE \) and \( CE \) are equal, implying the triangle is isosceles. - Point \( F \) is marked on \( AE \) with \( AF = 5 \). - Point \( D \) is marked on \( CE \). - \( EB \) is perpendicular to \( AC \) and line segment \( EB \) splits \( \Delta AEC \) into two right triangles. Analysis: - The centroid \( G \) divides each median into a 2:1 ratio, with the longer segment being between the vertex and the centroid. - \( EB \) is a perpendicular height from \( E \) to \( AC \). The goal of this problem is to find the length of \( EB \).
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