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Question
In ΔABC, D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of ΔADE and ΔABC.
Solution
It is given that D and E are midpoints of AB and AC.
Applying midpoint theorem, we can conclude that DE ‖ BC.
Hence, by B.P.T., we get :
`(AD)/(AB)=(AE)/(AC)`
Applying SAS similarity theorem, we can conclude that Δ ADE~ Δ ABC.
Therefore, the ration of areas of these triangles will be equal to the ratio of squares of their corresponding sides.
∴ `(9ar(ΔADE))/(ar(ΔABC))=(DE)^2/(BC)^2`
= `(1/2 BC)^2/(BC)^2`
= `1/4`
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