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The median and the height of an isosceles trapezoid are equal in length. One diagonal of the trapezoid has length 12. What is the area of the trapezoid?

 Apr 24, 2021
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See  the  following image  ( not to scale)

 

 

Call the length  of  the longer  base,  AB =  P  and the length  of  the  shorter base,  GF =  Q

 

Then  AF  =   Q  +  ( P  - Q)  / 2  =  (P + Q)  / 2   =   the  median length   =  M

 

So....since the  median  =  the  height, we  have  right  triangle AFD  such  that

 

AF^2  +  FD^2  =   AD^2          where  AF, FD =  M     and   AD   = 12

 

M^2  + M^2   =  12^2

 

2M^2  =  144

 

M^2  =  72

 

M  =  6sqrt (2)   =  height

 

So....the  area  =  (1/2)  height  * (sum of the bases)  =  height *  (sum  of bases)/2   =

 

height  *  median  =  

 

(6sqrt (2) )  ( 6sqrt (2))   =     36  * 2   =   72 units^2

 

cool cool  cool

 Apr 24, 2021

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