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In AABC, D and E are the midpoints of AB and CB. A) If DE = x + 9 and AC = 8x - 6, find x. B) If BE = 3y and EC = 5(y - 4), find y. E C

In AABC, D and E are the midpoints of AB and CB. A) If DE = x + 9 and AC = 8x - 6, find x. B) If BE = 3y and EC = 5(y - 4), find y. 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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In \( \triangle ABC \), \( D \) and \( E \) are the midpoints of \( \overline{AB} \) and \( \overline{CB} \). **A)** If \( DE = x + 9 \) and \( AC = 8x - 6 \), find \( x \). **B)** If \( BE = 3y \) and \( EC = 5(y - 4) \), find \( y \). **Diagram Explanation:** The diagram shows a triangle \( \triangle ABC \) with a line segment \( DE \) parallel to the base \( AC \). Points \( D \) and \( E \) are located on sides \( AB \) and \( CB \) respectively, indicating that \( DE \) is a midsegment connecting the midpoints of \( AB \) and \( CB \). This forms two smaller triangles, \( \triangle ADE \) and \( \triangle CBE \), within \( \triangle ABC \). The properties of midsegments suggest that \( DE \) is parallel to \( AC \) and \( DE = \frac{1}{2}AC \).
Transcribed Image Text:In \( \triangle ABC \), \( D \) and \( E \) are the midpoints of \( \overline{AB} \) and \( \overline{CB} \). **A)** If \( DE = x + 9 \) and \( AC = 8x - 6 \), find \( x \). **B)** If \( BE = 3y \) and \( EC = 5(y - 4) \), find \( y \). **Diagram Explanation:** The diagram shows a triangle \( \triangle ABC \) with a line segment \( DE \) parallel to the base \( AC \). Points \( D \) and \( E \) are located on sides \( AB \) and \( CB \) respectively, indicating that \( DE \) is a midsegment connecting the midpoints of \( AB \) and \( CB \). This forms two smaller triangles, \( \triangle ADE \) and \( \triangle CBE \), within \( \triangle ABC \). The properties of midsegments suggest that \( DE \) is parallel to \( AC \) and \( DE = \frac{1}{2}AC \).
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