Class 10

Math

All topics

Coordinate Geometry

$A(7,−3),B(5,3)$ and $C(3,−1)$ are the vertices of a $ΔABC$ and AD is its median. Prove that the median AD divides $ΔABC$ into two triangles of equal areas.

D is midpoint of BC, so find its coordinates using below:

Midpoint formula: $(x,y)=(2x_{1}+x_{2} ,2y_{1}+y_{2} )$

$D=((3+5)/2,(3−1)/2)=(4,1)$

Find area of triangle ABD:

We know that:

Area of a triangle $=21 [x_{1}(y_{2}−y_{3})+x_{2}(y_{3}−y_{1})+x_{3}(y_{1}−y_{2})]$

So,

Area of triangle ABD:

$=21 (7(3−1)+5(1+3)+4(−3−3))$

$=21 (14+20−24)$

$=21 (10)$

$=5$ sq. units .......$(1)$

Area of triangle ACD:

$=21 (7(−1−1)+3(1+3)+4(−3+1))$

$=21 (−14+12−8)$

$=21 (10)$

$=5$ sq. units ........$(2)$

From $(1)$ and $(2)$, we conclude that Area of triangle ABD and ACD is equal.

Hence proved.