In triangle PQR, the line ST is drawn parallel to QR so that PS= 3SQ. prove that PT = 3TR

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- Dec 31st 2008, 11:30 AMTweetyvector , geometry
In triangle PQR, the line ST is drawn parallel to QR so that PS= 3SQ. prove that PT = 3TR

- Jan 1st 2009, 09:57 AMearboth
1. Let $\displaystyle PT = k \cdot TR$

2. You are dealing with similar triangles. So use proportion:

$\displaystyle \dfrac{PS}{PS + SQ} = \dfrac{PT}{PT + TR} $

3. Substitute

$\displaystyle PS= 3 \cdot SQ$ ...... and ...... $\displaystyle PT = k \cdot TR$

$\displaystyle \dfrac{3 \cdot SQ}{3 \cdot SQ + SQ} = \dfrac{k \cdot TR}{k \cdot TR + TR} $

$\displaystyle \dfrac{3 \cdot SQ}{4 \cdot SQ } = \dfrac{k \cdot TR}{(k+1) \cdot TR } $

$\displaystyle \dfrac{3}{4 } = \dfrac{k}{(k+1) } ~\implies~3(k+1) = 4k ~\implies~\boxed{3=k}$ - Jan 1st 2009, 11:11 AMTweety
thanks so much for that, I would have never figured out how to prove it!