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

Here's the way I set it up, ACG :

Look at  triangle YGM

Note that  G  is the midpoint of XM...so  the height of G  above base YM = the height of YGM  =  (1/2)height of triangle XYZ

And since M is the midpoint of YZ, it is  the base of YGM  =  1/2 base of triangle XYZ

And triangles are to each other as the ratio of their bases and heights

So area of triangle YGM  = (1/2)* (1/2 base of XYZ) * (1/2 height of XYZ) =

(1/4) (1/2 base of XYZ) *(height of XYZ)  

(1/4)area of XYZ  =   1/4 XYZ

Now  ...look at triangle YTM

Since YG is  a median of triangle YXM....then YT  = 2/3  of YG

But....this means that T  is (2/3)  the height above base YM that G is

So...the height of triangle YTM  =  2/3   height of triangle YGM

And they are on the same base 

So...the areas of triangles on the same base are to each other as their heights....so..

Area of triangle YTM  = (2/3)area of triangle YGM  =  (2/3)(1/4)XYZ  = (1/6)XYZ

So....area  of triangle  MTG  = 

Area of triangle YGM - Area of triangle YTM =

(1/4)XYZ - (1/6)XYZ  =

(1/4  - 1/6) XYZ =

(2/24)XYZ  =

(1/12)XYZ =

(1/12)(150)  =

12.5 units^2