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 is a geometry question.
![### Geometry Proof: Proportional Segments in Triangles
#### Given:
\[
\overline{QT} \parallel \overline{RS}
\]
#### Prove:
\[
\frac{QU}{RV} = \frac{UT}{VS}
\]
#### Diagram Description:
- The diagram includes a triangle \(PRS\) with point \(P\) at the top, and points \(R\) and \(S\) at the bottom, forming the base.
- Line segment \(\overline{QT}\) is drawn parallel to line segment \(\overline{RS}\).
- Points \(Q\) and \(T\) lie on the sides \(\overline{PR}\) and \(\overline{PS}\) respectively.
- Points \(U\) and \(V\) are intersection points such that \(\overline{PV}\) is a segment within triangle \(PRS\).
#### Explanation:
Given that \(\overline{QT}\) is parallel to \(\overline{RS}\), by the Basic Proportionality Theorem (Thales' theorem), it is to be proven that the segments created on the sides of the triangle \(PRS\) are proportional:
\[
\frac{QU}{RV} = \frac{UT}{VS}
\]
This setup implies that if two lines are parallel and intersect two sides of a triangle, then they divide those sides proportionally.
This proof typically involves utilizing properties of similar triangles created by the parallel lines within the larger triangle \(PRS\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e8eb507-a403-445f-ba1a-c575542338e7%2Fcddbb981-156b-46ef-bb10-b10b648d10b0%2Frw72wve_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Proof: Proportional Segments in Triangles
#### Given:
\[
\overline{QT} \parallel \overline{RS}
\]
#### Prove:
\[
\frac{QU}{RV} = \frac{UT}{VS}
\]
#### Diagram Description:
- The diagram includes a triangle \(PRS\) with point \(P\) at the top, and points \(R\) and \(S\) at the bottom, forming the base.
- Line segment \(\overline{QT}\) is drawn parallel to line segment \(\overline{RS}\).
- Points \(Q\) and \(T\) lie on the sides \(\overline{PR}\) and \(\overline{PS}\) respectively.
- Points \(U\) and \(V\) are intersection points such that \(\overline{PV}\) is a segment within triangle \(PRS\).
#### Explanation:
Given that \(\overline{QT}\) is parallel to \(\overline{RS}\), by the Basic Proportionality Theorem (Thales' theorem), it is to be proven that the segments created on the sides of the triangle \(PRS\) are proportional:
\[
\frac{QU}{RV} = \frac{UT}{VS}
\]
This setup implies that if two lines are parallel and intersect two sides of a triangle, then they divide those sides proportionally.
This proof typically involves utilizing properties of similar triangles created by the parallel lines within the larger triangle \(PRS\).
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