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In the given ...

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Question 1
In the given figure, E is any point on the median AD of a $Δ A B C$ . Show that ar(ABE) = ar(ACE).

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Solution

AD is the median of $Δ A B C$ .
Therefore, it will divide $Δ A B C$ into two triangles of equal areas.
$∴$ Area $(Δ A B D) = A r e a (Δ A C D) . . . (1)$
ED is the median of $Δ E B C$ .
$∴$ Area $(Δ E B D) = A r e a (Δ E C D) . . . (2)$
On subtracting equation (2) from equation (1), we obtain,
$A r e a (Δ A B D) - A r e a (E B D) = A r e a (Δ A C D) - A r e a (Δ E C D)$
$A r e a (Δ A B E) = A r e a (Δ A C E)$

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