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# help geometry

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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 $AB = 20$, $AC = 20$, and BC = 30.

Apr 30, 2024

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AM =  sqrt [ 20^2 - 15^2]  =  sqrt [175 ]  =  5sqrt (7)

DM = AM / 2  =   2.5 sqrt (7)

Draw ED

Triangle AED ≈ Triangle ABM

AD / AM  = ED / BM

(1/2)AM / AM  = ED / BM

1/2  = ED / BM

ED = (1/2) BM

Triangle EDT ≈ BMT

Since ED = 1/2 BM

Then the height of triangle EDT  is 1/2 the height of triangle BMT

So the height of triangle BMT =  [ 2 /( 1 + 2)  ] * DM  = (2/3)DM

And the  height of triangle BME = DM

So  the area of triangle BME = (1/2)(BM)((DM)

And  the area  of triangle BMT = (1/2)(BM)(2/3)(DM)  = (1/3) (BM)(DM)

So  the area of triangle BET = [ BME ]  - [ BMT ]   =

(1/2)(BM)(DM)   -  (1/3) (BM)(DM) =  (1/6) (BM) (DM)  =

(1/6) (15) ( 2.5 sqrt (7) )  =

6.25 sqrt 7 =

25sqrt (7)  / 4

Apr 30, 2024