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Point D is the midpoint of median AM of triangle ABC. Point E is the midpoint of AB, and point T is the intersection of BD and ME. Find the area of triangle ADT if [DET] = 14.

 Jun 25, 2024
 #1
avatar+129725 
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Let TG be a median of triangle BMT

And triangle BMT is similar to triangle DET

So FT is a median of  triangle DET

Then the area of traingle FDT = (1/2) [ DET ]  = 7

 

And triangle AED  similar to triangle ABM

So ED  =   1/2  BM

And FD  = 1/2 GM

So the height of triangle FDT  = 1/2 height of triangle GTM

So FT / FG =  FT / [ FT + 2FT] = 1/3

So FT = (1/3)FG 

FG = 3FT

And FG = AF

So AF = 3 FT

 

So  the height of triangle AFD  = 3 times the height of triangle FDT

And  since they're on the same  base (ED)..... the area of triangle AFD =  3 [ FDT]  =  3 * 7 = 21

 

[ ADT ] = [ AFD ] + [ FDT ] =  21 + 7  =  28 

 

cool cool cool

 Jun 25, 2024

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