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# help geometry

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Quadrilateral PQRS is a trapezoid with bases PQ and RS. The median MN meets the diagonals PQ and RS at X and Y, respectively. If SR=20 and XY=5, find PQ.

Feb 11, 2021

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Let Z  be  the intersection point  of the diagonals

And  note  that triangle  XYZ  is similar to triangle RZS

So since  RS =  20  and XY  = 5, the  height of triangle RZS  is  4 times  that of  the height of  triangle XYZ

So....there are 5 equal parts  of the  distance  from SR to the midline

And one  of these parts is the  height of triangle XYZ

And 4 of the parts  = the height of triangle RZS

And  triangle  PZQ  is similar to triangle  RZS

And  there  are  also  5 of the same  equal parts from PQ  to the  midline

And so  there  are  6 equal  parts  to  the  height  of  triangle PZQ....the 5equal parts to  the midline and the one other equal part  of the height of triangle XYZ

So.....the  height of  triangle  PZQ  to the height of triangle RZS =  6/4 = 3/2

So....the  base of  triangle PZQ = (3/2)  the  base of triangle RZS =  (3/2) 20  = 30  = PQ

Feb 12, 2021