1.

In a triangle ABC, E is the midpoint of median AD. Show that ar ΔABE = ar ΔACE

Answer»

In ΔABC; AD is a median. 

∴ ΔABD = ΔACD …………….. (1) 

(∵ Median divides a triangle in two equal triangles)

Also in ΔABD; BE is a median. 

∴ ΔABE = ΔBED = 1/2 ΔABD …………..(2)

Also in ΔACD; CE is a median.

∴ ΔACE = ΔCDE = 1/2 ΔACD …………….(3)

From (1), (2) and (3);

ΔABE = ΔACE

(OR)

ΔABD = ΔACD [∵ AD is median in ΔABC]

1/2 ΔABD = 1/2 ΔACD

[Dividing both sides by 2] 

ΔABE = ΔAEC 

[∵ BE is median of ΔABD, CE is median of ΔACD] 

Hence proved.



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