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Question
In figure, if DE || BC, find the ratio of ar(ADE) and ar(DECB).
Solution
Given,
DE || BC,
DE = 6 cm
And BC = 12 cm
In ΔABC and ΔADE,
∠ABC = ∠ADE ...[Corresponding angle]
∠ACB = ∠AED ...[Corresponding angle]
And ∠A = ∠A ...[Common side]
∴ ΔABC ∼ ΔAED ...[By AAA similarity criterion]
Then, `("ar(ΔADE)")/("ar(ΔABC)") = ("DE")^2/("BC")^2`
= `(6)^2/(12)^2`
= `(1/2)^2`
⇒ `("ar(ΔADE)")/("ar(ΔABC)") = (1/2)^2 = 1/4`
Let ar(ΔADE) = k,
Then ar(ΔABC) = 4k
Now, ar(DECB) = ar(ABC) – ar(ΔADE)
= 4k – k
= 3k
∴ Required ratio = ar(ADE) : ar(DECB)
= k : 3k
= 1 : 3
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