Question

In the given figure, \(AB \parallel DE\) and \(AC \parallel DF\). Show that \(\triangle ABC \sim \triangle DEF\). If \(BC = 10\) cm, \(EB = CF = 5\) cm and \(AB = 7\) cm, then find the length DE.

Hint:

When proving similarity with parallel lines, look for the 'F' shape for corresponding angles. Always ensure you add up the segments correctly to find the full length of the side of the larger triangle.

Answer 1

Step 1: Proving Similarity of Triangles:
Given:
AB ∥ DE
AC ∥ DF

Since EF acts as a transversal:

∠ABC = ∠DEF (Corresponding angles, AB ∥ DE)
∠ACB = ∠DFE (Corresponding angles, AC ∥ DF)

Thus two corresponding angles are equal.

Therefore,
ΔABC ∼ ΔDEF (AA similarity criterion)

Step 2: Using Proportionality of Similar Triangles:
For similar triangles, corresponding sides are proportional.

AB / DE = BC / EF

Given:
AB = 7 cm
BC = 10 cm
EB = 5 cm
CF = 5 cm

Total EF = EB + BC + CF
= 5 + 10 + 5
= 20 cm

Step 3: Substituting Values:
7 / DE = 10 / 20
7 / DE = 1 / 2

Cross multiplying:
DE = 7 × 2
DE = 14 cm

Final Answer:
The length of DE is 14 cm.