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# geometry

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Point G is the midpoint of median line XM of triangle XYZ. Point H is the midpoint of line XY, and point T is the intersection of line HM and line YG. Find the area of triangle MTG if [XYZ] =250.

Feb 17, 2022

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+1 Triangle XYZ  is isosceles   (XZ = YZ)  with a base of 25   and a height of 20

[XYZ]  = (1/2) ( 25 ) ( 20)  =  250

Note  that in triangle   XYM, GY and  HM are medians

And triangle  HYM   is similar to triangle XYZ  because HY  = XY/2  and MY = ZX / 2  = ZY/2

And the scale factor of  triangle HYM  to triangle XYZ =  (1/2)

So  the area of triangle  HYM =  (scale factor )^2  * ( area of XYZ)  =  (1/4)(250) =  125/2

And the areas of triangles  HYM , XHM and XYG are the same

So T is  the intersection of  medians GY and HM in triangle XM

And by the property of intersecting medians,   HT =  1/3 of HM

So, since they are on the  same base, triangle HYT = (1/3)area of triangle HYM =  (125/2) /3 =  125/6

Since [ XYG ] =  125/2

Then  [XGTH]  has an area of  (2/3) (125/2)   = 125 / 3

But since triangles XHM and XYG have the same areas and XGTH is common to each, then the area of triangle MTG =  [ XHM ]  - [ XGTH]  =   125/2  - 125/3 =   125 / 6   Feb 17, 2022