Question

What is the ratio between length of perpendicular AO and the perimeter of the $△$ABC?

• A. 1 : 2
• B. 1 : $2\sqrt{2}$
• C. 1 : 2(1 + $\sqrt{2})$
• D. 1 : (2 + $\sqrt{2}$)

Answer 1

The correct option is C

1 : 2(1 + $\sqrt{2})$

$△$ ABC is an isosceles triangle.

$\angle$ ABO = ${45}^{\circ }$

$\therefore$ $△$ AOB is a 45-45-90 triangle and ratio of their sides = $1:1:\sqrt{2}$

Let $AO=OB=x$ and $AB=\sqrt{2}x$

$AB=2\sqrt{2}=\sqrt{2}x$

$\Rightarrow x=2$

$\therefore AO=OB=2$

Similarly, $AO=OC=2$

$\therefore$ The length of BC = 2 + 2 = 4 ($\because$ BC = BO + OC)

$\therefore$ Perimeter of the $△$ ABC
= AB + BC + CA
= $2\sqrt{2}+4+2\sqrt{2}=4+4\sqrt{2}$

Ratio Between AO and perimeter $=2:4+4\sqrt{2}=1:2(1+\sqrt{2})$