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To PROVE THAT : AC + AD = BC
Since AB = AC = a , <A = 90°
=> BC² = a² + a² = 2a² ( by Pythagoras law)
=> BC = √2a
Now, since CD is bisector of < C
=> AC/ BC = AD/BD ( by angle bisector theorem)
=> a/√2a = AD/BD
=> 1 /√2 = AD/BD = x/√2x
=> AB = x + √2x = x(1+√2)
So, AC = x(1+√2)
By Pythagoras law
AB² + AC² = BC²
=> x²(1+√2)² + x² (1+√2)² = BC²
=> BC = √{2x²(1+√2)²}
=> BC = √2 x ( 1+√2)
=> BC = √2x + 2x ………….. (1) (RHS)
Now, since AC = x + √2x
& AD = x
=> AC + AD = x+ √2x + x
=> AC + AD = √2x + 2x ………… ( 2) LHS
therefore LHS=RHS
I hope this will help you.
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