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if the side AB~PQ , AC~PR and median AD~PM it means it all side are proportional to one
another then the triangle ABC~PQR
another then the triangle ABC~PQR
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in my book the answer is till here only is it right
ur answer is correct
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We can use Apollonius' theorem here.
Median AD through A on to BC:
AD = √[ (2AC²+ 2AB² - BC²)/4 ]
=> BC² = 2 AC² + 2 AB² - 4 AD² --- (1)
Given also: in ΔPQR
QR² = 2 PQ² + 2 PR² - 4 PM² --- (2)
We are given AB/PQ = AC/PR = AD/PM = k (say)
So AB =k PQ , AC = k PR, AD = k PM.
Substitute these in (1) to get:
BC² = 2 k² PR² + 2 k² PQ² - 4 k² PM²
= k² * QR²
=> BC = k QR.
Since AB/PQ = AC/PR = BC/QR, the two triangles are similar.
Proved.
Median AD through A on to BC:
AD = √[ (2AC²+ 2AB² - BC²)/4 ]
=> BC² = 2 AC² + 2 AB² - 4 AD² --- (1)
Given also: in ΔPQR
QR² = 2 PQ² + 2 PR² - 4 PM² --- (2)
We are given AB/PQ = AC/PR = AD/PM = k (say)
So AB =k PQ , AC = k PR, AD = k PM.
Substitute these in (1) to get:
BC² = 2 k² PR² + 2 k² PQ² - 4 k² PM²
= k² * QR²
=> BC = k QR.
Since AB/PQ = AC/PR = BC/QR, the two triangles are similar.
Proved.
:-)
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