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] =150.
I got some help on this earlier but I'm still in the dark. Thanks!
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 =