Question

∆ABC and ∆DEF are equilateral triangles. If A(∆ABC) : A(∆DEF) = 1 : 2 and AB = 4, find DE.

Answer 1

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