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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.
Solution
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.
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