In $Δ A B C$ . and E are the mid-points of AB and AC respectively. Find the ratio of the areas of $Δ A E D$ and $Δ A B C$ .

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Solution

In ΔADE and ΔABC
∠C = ∠E [DE||BC]
∠A = ∠A [common]
Then, ΔADE ~ ΔABC [By AA similarity]
$s o space fraction numerator a r left parenthesis A E D right parenthesis over denominator a r left parenthesis A B C right parenthesis end fraction space equals space open parentheses fraction numerator A D over denominator A B end fraction close parentheses squared sin c e space A D thin space equals space 1 half A B h e n c e space fraction numerator A D over denominator A B end fraction space equals space 1 half fraction numerator a r left parenthesis A E D right parenthesis over denominator a r left parenthesis A B C right parenthesis end fraction space equals open parentheses 1 half close parentheses squared equals space 1 fourth$

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