If two sides and median bisecting one of these side of a triangle are equal respectively to the two sides and the median bisecting one of these sides of another triangle, prove that two triangles are congruent.
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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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