Question

i) A ABC and A DEF are equilateral triangles,
A( A ABC) : A(A DEF) - 1:2. IT AB=4 then what is length of DE?​

Answer 1

Corrrect Question:

$\red\bigstar\triangle$ ABC and $\triangle$ DEF are equilateral triangles. A( $\triangle$ ABC) : A($\triangle$ DEF) = 1:2. If AB=4 then what is length of DE?

Given:

• A ( $\triangle$ ABC) : A ( $\triangle$ DEF) = 1:2
• AB = 4

Find:

• Length of DE

Solution:

we, know that

All equilateral Triangles are similar to each other.

And, if two triangles are similar then the ratio of the area of Triangles is equal to square to the ratio of its corresponding sides.

So,

$\green\star \sf \dfrac{ A( \triangle ABC) }{A( \triangle DE F)} = {\bigg( \dfrac{ AB }{ DE }\bigg)}^{2}$

where,

• A ( $\triangle$ ABC) : A ( $\triangle$ DEF) = 1:2
• AB = 4

SO,

$\dashrightarrow \sf \dfrac{ A( \triangle ABC) }{A( \triangle DE F)} = {\bigg( \dfrac{ AB }{ DE } \bigg) }^{2}$

$\pink\dashrightarrow \sf \dfrac{1}{2} = {\bigg( \dfrac{4}{ DE } \bigg) }^{2}$

$\pink\dashrightarrow \sf \dfrac{1}{2} = \dfrac{16}{ {DE}^{2} }$

$\qquad\quad$ Cross-multiplication

$\pink\dashrightarrow \sf \dfrac{1}{2} = \dfrac{16}{ {DE}^{2} }$

$\pink\dashrightarrow \sf 16 \times 2 = {DE}^{2} \times 1$

$\pink\dashrightarrow \sf 32= {DE}^{2}$

$\pink\dashrightarrow \sf \sqrt{32}= DE$

$\pink\dashrightarrow \sf DE = \sqrt{32}$

$\pink\dashrightarrow \sf DE = \sqrt{16 \times 2}$

$\pink\dashrightarrow \sf DE = 4\sqrt{2}$

Hence, Length of DE = 4√2