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Question
In ΔABC, D and E are the mid-points of AB and AC respectively. Find the ratio of the areas of ΔADE and ΔABC
Solution
We have, D and E as the mid-points of AB and AC
So, according to the mid-point theorem
DE || BC and DE `=1/2`BC ...(i)
In ΔADE and ΔABC
∠A = ∠A [Common]
∠ADE = ∠B [Corresponding angles]
Then, ΔADE ~ ΔABC [By AA similarity]
By area of similar triangle theorem
`("Area"(triangleADE))/("Area"(triangleABC))="DE"^2/"BC"^2`
`=(1/2BC)^2/"BC"^2` [From (i)]
`=(1/4BC^2)/"BC"^2`
`=1/4`
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