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Question
∆ABC and ∆DEF are equilateral triangles. If A(∆ABC) : A(∆DEF) = 1 : 2 and AB = 4, find DE.
Solution
In ∆ABC and ∆DEF,
`{:(∠"A" ≅ ∠"D"),(∠"B" ≅ ∠"E"):} ...}("Each angle is of measure 60°")`
∴ ∆ABC ∼ ∆DEF ...(AA test of similarity)
By the Theorem of areas of similar triangles,
∴ `"A(∆ABC)"/"A(∆DEF)" = "AB"^2/"DE"^2`
∴ `1/2 = 4^2/"DE"^2`
∴ DE2 = 42 × 2
Taking square root of both sides,
∴ DE = 4√2 units
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