Question

In the given figure, AB || DE and AC || DF. Show that ΔABC ~ ΔDEF. If BC = 10 cm, EB = CF = 5 cm and AB = 7 cm, then find the length DE.

Answer 1

Given:

AB || DE

AC || DF

BC = 10 cm

EB = CF = 5 cm

AB = 7 cm

We need to show that ΔABC ~ ΔDEF and then find the length of DE.

Step 1: Show that ΔABC ~ ΔDEF

Since AB || DE and AC || DF, corresponding angles are equal.

∠BAC = ∠EDF   ...(Corresponding angles)

∠ABC = ∠DEF   ...(Corresponding angles)

Also, ∠ACB = ∠DFE   ...(Corresponding angles)

Since all corresponding angles are equal, by AA (Angle-Angle) similarity criterion,

ΔABC ~ ΔDEF

Step 2: Find the length of DE

From the figure and given data:

BC = 10 cm

EB = 5 cm

CF = 5 cm

Since EB = CF = 5 cm. 

The length EF = EB + BC + CF

= 5 + 10 + 5

= 20 cm

Now, since △ABC ∼ △DEF, corresponding sides are proportional:

`(AB)/(DE) = (BC)/(EF) = (AC)/(DF)`

We know:

AB = 7 cm

BC = 10 cm

EF = 20 cm

Using the proportion:

`(AB)/(DE) = (BC)/(EF)`

⇒ `7/(DE) = 10/20`

⇒ `7/(DE) = 1/2`

Cross-multiplying:

7 × 2 = DE

⇒ DE = 14 cm