Question

If triangle ABC~DEF , Find AB/DE.​

Answer 1

(i) Consider the quadrilateral ABED

We have , AB=DE and AB∥DE

One pair of opposite sides are equal and parallel. Therefore

ABED is a parallelogram.

(ii) In quadrilateral BEFC , we have

BC=EF and BC∥EF. One pair of opposite sides are equal and parallel.therefore ,BEFC is a parallelogram.

(iii) AD=BE and AD∥BE ∣ As ABED is a ||gm ... (1)

and CF=BE and CF∥BE ∣ As BEFC is a ||gm ... (2)

From (1) and (2), it can be inferred

AD=CF and AD∥CF

(iv) AD=CF and AD∥CF

One pair of opposite sides are equal and parallel

⇒ ACFD is a parallelogram.

(v) Since ACFD is parallelogram.

AC=DF ∣ As Opposite sides of a|| gm ACFD

Answer 2

There's a mistake of labeling in the figure. The Correct figure is given in the attachment.

$\:$

Solution:

It is given that ΔABC ∼ ΔDEF

So their corresponding sides will be proportional.

$\implies \sf{\dfrac{AB}{DE}=\dfrac{BC}{EF}=\dfrac{AC}{DF}}$

Out of these let's consider 2 pairs.

$\leadsto \sf{\dfrac{AB}{DE}=\dfrac{AC}{DF}}$

Substitute the values.

$\implies \sf{\dfrac{2x-1}{18}=\dfrac{3x}{6x}}$

Further simplifying,

$\implies \sf{\dfrac{2x-1}{18}=\dfrac{1}{2}}$

On cross multiplication,

$\implies \sf{2(2x-1)=18}$

$\implies \sf{4x-2=18}$

$\implies \sf{4x=20}$

$\implies \sf{x = \dfrac{20}{4}}$

$\implies \boxed{\boldsymbol{x=5}}$

Now let's find out AB/DE

$\sf{\dfrac{AB}{DE}=\dfrac{2x-1}{18}}$

Put the value of x here.

$\dashrightarrow \: \: \sf{\dfrac{AB}{DE}=\dfrac{2(5)-1}{18}}$

$\dashrightarrow \: \: \sf{\dfrac{AB}{DE}=\dfrac{10-1}{18}}$

$\dashrightarrow \: \: \sf{\dfrac{AB}{DE}=\dfrac{9}{18}}$

$\therefore \boxed{\boldsymbol{\dfrac{AB}{DE}=\dfrac{1}{2}}}$