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Question

In right-angled triangle ABC in which C=90, If D is the mid-point of BC, prove that AB2=4AD23AC2.

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Solution

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In ACD

AD2=AC2+CD2 .....(1)

In ACB

AB2AC2=BC2 .....(2)

As CD=DB=BC2 ...... (3)

Substituting (3) in (1)
AD2=AC2+BC42

4AD24AC2=BC2 ....... (4)

Subtracting (2) from (4)

4AD24AC2(AB2AC2)=BC2BC2

4AD24AC2AB2+AC2=0

4AD23AC2=AB2

Hence proved.

893402_969508_ans_3863a8d0be134c8fac6fbe07ad0223eb.png

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