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!

AnonymousConfusedGuy
Mar 9, 2018

#1**+2 **

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

CPhill
Mar 9, 2018