Answer:

Given:  In ∆ ABC and ∆PQR ,AD and PM  are their medians.

AB/PQ = AC/PR= AD/PM…….(1)

TO PROVE:

∆ABC~∆PQR

Construction;

Produce AD to E  such that AD=DE &  produce PM to N such that PM= MN.

join BE,CE,QN,RN

PROOF:

Quadrilaterals ABEC and PQNR are parallelograms because their diagonals bisect each other at D and M.

BE= AC & QN= PR

BE/AC=1 & QN/PR=1

BE/AC =QN/PR or BE/QN = AC/PR

BE/QN= AB/PQ   [ From eq1]

or AB/PQ= BE/QN…….(2)

From eq 1

AB/PQ= AD/PM= 2AD/2PM= AE/PN

[SINCE DIAGONALS BISECT EACH OTHER]

AB/PQ= AE/PN…………..(3)

From equation 2 and 3

AB/PQ=BE/QN= AE/PN

∆ABE ~∆PQN

∠1= ∠2…………..(4)

[Since corresponding angles of two similar triangles are equal]

Similarly we can prove that

∆ACE ~∆PRN

∠3=∠4…………(5)

ON ADDING EQUATION 4 AND 5

∠1+∠2=∠3+∠4

∠BAC = ∠QPR

and AB/PQ= AC/PR   [from equation 1]

∆ABC~∆PQR

[By SAS similarity criteria]

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