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Question

In the given figure, ADBC and BD=13CD. Prove that 2CA2=2AB2+BC2.
1390082_8998c050522e4330b704b81962adb0cf.png

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Solution

Given ABC is a triangle and ADBC.

To prove : 2CA22.AB2+BC2

proof : BD+CD=BC

13.CD+CD=BC

CD+3CD=3.BC

4CD=3BC

CD=34.BC(1)

From (1)
BD=34.BC

BD=BC4(2)

In ACD,
AC2=AD2+DC2(3)

In ADB,
AB2=AD2+BD2
AD2=AB2BD2(4)

put (4) in (1)

AC2=AB2BD2+DC2

AC2=AB2(BC4)2+(34.BC)2

AC2=AB2BC216+9.BC216

AC2=AB2+BC2+9BC216

AC2=AB2+8.BC216

AC2=AC2=2.AB2+BC22

2AC2=2AB2+BC2

Hence it proved.

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