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.

Guest Feb 17, 2022

#1**+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

CPhill Feb 17, 2022