Question

In the figure with ΔABC, P, Q, R are the mid-points of AB, AC and BC respectively. Then prove that the four triangles formed are congruent to each other.

Answer 1

Given: ABC is a triangle and P, Q and R are the midpoints of sides AB, AC, and BC, respectively.

To prove: ΔABC is divided into 4 congruent triangles.

Proof: P and Q are the mid-points of AB and AC of ΔABC.

So, PQ || BC  ......[Line segment joining the mid-points of two sides of a triangle is parallel to the third side]

Similarly, PR || AC and QR || AB

In PQRB, PB || QR and PQ || BR  ......[Parts of parallel lines are parallel]

i.e., Both opposite pairs of sides are parallel.

As a result, PQRB is a parallelogram.

As a result, PR is a diagonal of the parallelogram PQRB.

So, ΔPBR ≅ ΔPQR  .....[A diagonal of a parallelogram divides it into two congruent triangles.] .....(i)

APRQ is also a parallelogram.

So, ΔAPQ ≅ ΔPQR  .....(ii)

PQRC is also a parallelogram.

So, ΔPQR ≅ ΔQRC  ......(iii)

From equations (i), (ii) and (iii),

ΔPBR ≅ ΔPQR ≅ ΔAPQ ≅ ΔQRC

As a result, all four triangles are congruent.

Hence Proved.