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Question

ABC is a right triangle right angled at A such that AB=AC and bisector of C intersects the side AB at D. prove that AC+AD=BC.

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Solution

Let AB=AC=a and AD=b
In a right angled triangle ABC,BC2=AB2+AC2
BC2=a2+a2=2a2
BC=a2

Given AD=b, we get
DB=ABAD=ab

We have to prove that AC+AD=BC or a+b=a2

By the angle bisector theorem, we get

ADDB=ACBC

bab=aa2=12

b2=ab

b(2+1)=a

b=a2+1

b=a2+1×2121

b=a(21)

a+b=a2

or AD+AC=BC

Hence proved.

1261229_1179396_ans_e0b789cba57b4194aa9630a2056ccf88.PNG

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