∆ABC and ∆DEF are equilateral triangles, A(∆ABC): A(∆DEF) = 1: 2. If AB = 4 then what is length of DE? 2√2 4 8 4√2
Problem Set 1 | Q 1.4 | Page 26
∆ABC and ∆DEF are equilateral triangles, A(∆ABC): A(∆DEF) = 1: 2. If AB = 4 then what is length of DE?
2√2
4
8
4√2
∆ABC and ∆DEF are equilateral triangles. ...(Given)
In ∆ABC and ∆DEF,
`{:(∠"B" ≅ ∠"E"),(∠"A" ≅ ∠"D"):} ...("Measure of equilateral triangles is 60°")`
∴ ∆ABC ~ ∆DEF ...(By 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`
Taking square root both sides,
∴ `1/sqrt2 = 4/"DE"`
∴ DE = 4√2 units
Answer:-
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths. Since ∆ABC and ∆DEF are equilateral triangles, their areas can be expressed as A(∆ABC) = (√3/4)AB^2 and A(∆DEF) = (√3/4)DE^2, respectively.
Thus, we have:
A(∆ABC) : A(∆DEF) = 1 : 2
(√3/4)AB^2 : (√3/4)DE^2 = 1 : 2
AB^2 : DE^2 = 1 : 2 (canceling (√3/4) from both sides)
AB = 4, so AB^2 = 16
DE^2 = 2(AB^2) = 2(16) = 32
DE = √32 = √(16 × 2) = √16 × √2 = 4√2
Therefore, the length of DE is 4√2. Option (d) is the correct answer.
Problem Set 1 | Q 1.4 | Page 26