ABC is a right angled triangle such that AB=AC and bisector of angle c intersects the side AB at D. Prove that AC + AD =BC
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Let AB=AC=a and AD=b.
△ABC is right angled at A ⇒ AB
2
+AC
2
=BC
2
⇒BC=a
2
BD=AB−AD=a−b
By angle-bisector theorem,
BD
AD
=
BC
AC
.
⇒
a−b
b
=
a
2
a
=
2
1
⇒b=
1+
2
a
=a(
2
−1)
⇒a+b=a
2
But, we know AC=a, AD=b, BC=a
2
⇒AC+AD=BC. Hence it is proved
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