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# Q.let s be a square of unit area.consider a quadilateral which has 1 vertex on each side of the square ,if a,b,c,d denote the length of the side of the quadilateral ...P.T. 2<a^1+b^2+c^2+d^2<4 Askiitians Expert Soumyajit IIT-Kharagpur
28 Points
11 years ago

Dear Aman,

Ans:- Let the square be ABCD and the corresponding points be PQRS( P on AB Q on BC etc)

then we get the following eq

BP^2+BQ^2=a^2

AP^2+AS^2=b^2

SD^2+DR^2=c^2

QC^2+CR^2=d^2

AP^2+BP^2+BQ^2+QC^2+CR^2+RD^2+SD^2+AS^2=a^2+b^2+c^2+d^2 =X(let).................(1)

breaking eq (1) once in a^2+b^2=(a+b)^2 - 2ab form and again (a-b)^2+2ab form we get

X=(AP+BP)^2+(BQ+QC)^2+(CR+RD)^2+(SD+AS)^2 -2∑(AP.PB)

=4 - 2∑(AP PB)

X<4

AGAIN

X=∑(AP - PB)^2 +2∑(AP PB)

Hence X>2

So WE GET

2<X<4(Proved)

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