# Circle O with diameter "AB"=6 has point "C" on its circumference. If "AC"⋅"BC"="AB", then find the perimeter of triangle "ABC"?

## This is a tricky question.I tried to my best but could not solve it Please help, Thank you

Apr 9, 2017

perimeter $= 6 + 4 \sqrt{3} \approx 12.928$

#### Explanation:

Given that $A B$ is the diameter, $\implies \angle A C B = {90}^{\circ}$
$\implies A {B}^{2} = A {C}^{2} + B {C}^{2}$
$\implies 36 = A {C}^{2} + B {C}^{2}$

${\left(A C + B C\right)}^{2} = A {C}^{2} + B {C}^{2} + 2 \cdot A C \cdot B C$
Given $A C \cdot B C = 6$
$\implies {\left(A C + B C\right)}^{2} = 36 + 2 \times 6 = 48$
$\implies A C + B C = \sqrt{48} = 4 \sqrt{3}$

perimeter of $\Delta A B C = A B + A C + B C$
$= 6 + 4 \sqrt{3} \approx 12.928$