Question

In the given figure, ABC is a triangle in which L is the mid-point of BC & M is the mid-point of AL.

Prove that : ar (ΔAMC) = $\frac{1}{4}$ ar (ΔABC)

Explain with complete calculations & justifications.

Points : 15 ☺

Answer 1

In triangle ABC, AL is the median therefore ABL =ALC
And M is the midpoint of AL
So ML is 1/2 of triangle ALC
so ALC =1/2(1/2 ABC)
Therefore AMC =1/4 ABC

Answer 2

we know,
median divides a triangle to two triangles of equal area .

In triangle ABC,

midpoint of BC is L
And median is AL
therefore,
area of triangle ABL = area of triangle ACL
area of triangle ACL
= 1/2 x area of triangle ABC ---------(1)

In triangle ACL

midpoint of AL is M
And median CM
therefore,
area of triangle ACM = area of triangle CML
area of triangle ACM
= 1/2 x area of ACL
= 1/2 ( 1/2 x area of Triangle ABC)
[using (1)]
= 1/4 area of triangle ABC
(proved)