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Question

Let ΔPQR be a triangle. Let a=QR,b=RP. If |a|=12,|b|=43 and b.c=24, then which of the following is (are) true?

A
|c|22|a|=12
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B
|c|22+|a|=30
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C
|a×b+c×a|=483
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D
a.b=72
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Solution

The correct options are
A |c|22|a|=12
B |a×b+c×a|=483
D a.b=72
Using vectors summation method,
a=b+c
|a|2=|c|2+|b|22|c||b|cosP
|a|2=|c|2+|b|2+2|c||b|cos(πP)
|a|2=|c|2+|b|2+2cb
|a|2=122(43)22×24
c=43
|c|22|a|=12 ............... Option A.
c=a+b
|c|2=|a|2+|b|22|a||b|cosR
(|a||b|cosR=|a||b|cos(πR)=ab)
|c|2=122+(43)2+2ab
ab=72 ................. Option D.
ab=72|a||b|cos(πR)=72R=2π3
|b|=|c|=43
PQR is a isosceles triangle, with P=Q=π6
Now,
|a×b+c×a|=|a||b|sin(ππ6)+|c||a|sin(ππ6)
|a×b+c×a|=483 ............. option C.

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