Question

Prove that the four triangles formed by joining in pairs the mid-points of the sides C of a triangle are congruent to each other. [4 marks]

Answer 1

It is given that

In Δ ABC

D, E and F are the mid-points of AB, BC and CA

Now join DE, EF and FD

To find:

Δ ADF ≅ Δ DBE ≅ Δ ECF ≅ Δ DEF

To prove:

In Δ ABC

D and E are the mid-points of AB and BC

DE || AC or FC

Similarly DF || EC

DECF is a parallelogram (1 Mark)

We know that

Diagonal FE divides the parallelogram DECF in two congruent triangles DEF and CEF

Δ DEF ≅ Δ ECF …… (1) (1 Mark)

Here we can prove that

Δ DBE ≅ Δ DEF …. (2)

Δ DEF ≅ Δ ADF ……. (3)

Using equation (1), (2) and (3)

Δ ADF ≅ Δ DBE ≅ Δ ECF ≅ Δ DEF (2 Mark)