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Let ABCD be a trapezoid with midsegment MN, and suppose the diagonals split MN into three segments MP, PQ, and QN. If AB=8 and CD = 14, find the lengths of the midsegment MN and the three segments MP, PQ, and QN.

 Jul 20, 2022
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See the image below :

 

 

 

The length of the midsegment = (sum of bases ) / 2  =  (14 + 8)  / 2  =   11

For convenience, let the height of the trapezoid =  8    (it could be any height ....it works out the same)

 

Note that  DEC is similar to triangle BEA

Therefore  the height of triangle DEC =  (8/14)  =  of triangle BEA

So...the height of triangle  DEC  =  8 / (8 + 14) * (8)  =   (8/22)(8)  = 64/22 = 32/11

So  the height of triangle QEP =  4  - (32/11)  = 12/11 

And triangles DEC and QEP are similar

So  PQ =  (12/32)(8)  = 96/32  =  3

 

And, by symmetery , PQ = QN =  (11 - PQ) / 2  =   (11 - 3) / 2   =  8/2 =  4

 

 

cool cool cool

 Jul 20, 2022

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