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Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$.  Find the area of triangle $BET$ if $[DTM] = 20$.

 Jan 3, 2024
 #1
avatar+129245 
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Let

A= (a,b)

B=(0,0)

M = (c,0)

D = [ (a +c)/2 , b/2)

E = (a/2,b/2)

 

 

DE is  parallel to BM   because the slope of  both segments  =  0

 

BM = c

DE = c/2

So DE = (1/2)  of BM

 

Angle DET = Angle BMT

Angle DTE  = Angle BTM

 

So triangle EDT  is similar to  triangle MBT

 

Since DE =(1/2) of BM.....then  ET = (1/2)MT

 

Since triangles DET and DMT are under the same  height and ET = (1/2)TM  then [DET] = (1/2)[DMT ] = (1/2)(20) = 10

 

And DT = (1/2)BT  ⇒  2DT = BT

And since triangles DET and BET  are under the same height, then  [BET] = 2 [ DET] = 2 (10)   = 20

 

 

cool cool cool

 Jan 3, 2024

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