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In the diagram below, SR = PQ = 20 and RT:TS = 2:5. Find UV.

 

 Aug 25, 2024
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\(\angle{P} = \angle{TUV}\)

\(\angle{PUV} = \angle{UTR}\)

\(\angle{PQV} = \angle{UVR}\)

\(\angle{QVU} = \angle{VRT}\)

 

Therefore, trapezoid PUVQ is similar to trapezoid UTRV

 

\(\overline{TR} = \frac{2}{7}(20) = \frac{40}{7}\)

 

Since \(PUVQ \sim UTRV\):

 

\(\frac{20}{UV} = \frac{UV}{\frac{40}{7}}\)

\(\frac{800}{7} = \overline{UV}^2\)

\(\textbf{UV = }\frac{\textbf{20}\sqrt{\textbf{2}}}{\sqrt{\textbf{7}}}\)

\(\textbf{UV = }\frac{\textbf{20}\sqrt{\textbf{14}}}{{\textbf{7}}}\)

 

Let me know if I made any mistakes!

 Aug 26, 2024

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