Question

prove that the sum of the squares of two sides of a triangle is equal to twice the square of the median on the third side class have the square of the third side​

Answer 1

Theorem:- The sum of the square of two of a triangle is equal to twice the square on hall the third side plus 

twice the square on the median which bisects the 

third side.

Proof:- Given △ABC and AD is a median

We need to proof AB²+AC²=2AD²+2(21BC)²

i.e, AB²+AC²=2AD²+2BD².........(i)

Let AN⊥BC 

Then ln△ABN,AB2l²=AN²+BN²

ln △ANC,AC²=AN²+NC²...........(ii)

Add (i) and (ii)

We get  AB²+AC²=AN²+BN²+AN²+NC²

=2AN²+BN²+(DC−DN)²

=2AN²+(BD+DN)²+(DC−DN)²

=2AN²+BD²+DN²+2.BD.DN+DC²+DN²−2DC.DN

=2AN2+2DN2+BD2+DC2−2DC.DN+2BD.DN

=2(AN²+DN²)+BD²+BD²−2DC.DN+2BD.DN

=[∵BD=DC]

∴AB²+AC²=2AD²+2BD²

AB²+AC²=2AD²+2(1/2BC)²

AB²+AC²=2AD²+BC²

Hope this helps you

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