Question

Two perpendicular chords AB and CD of a circle intersect at O, (inside the circle AB is bisected at O). If AO = 4 units and OD = 6 units. Find circumradius of $\Delta ACO$.

• A. $\sqrt{13}$
• B. $\frac{2\sqrt{13}}{3}$
• C. $\frac{\sqrt{15}}{2}$
• D. $\frac{2\sqrt{15}}{4}$

Answer 1

The correct option is B $\frac{2\sqrt{13}}{3}$

When 2 chords intersect, the product of segment of one chord is equal to the product of segments of the other

$⟹AO\times BO=OC\times OD$

Given $AO=BO=4$

$⟹4\times 4=OC\times 6$

$OC=\frac{8}{3}$

Given $AB$ is perpendicular to $CD$ $⟹\angle COA=90$

$A{C}^2=O{C}^2+O{A}^2=\frac{64}{9}+16$

$AC=\frac{\sqrt{208}}{3}$

Since $AOC$ is a right triangle

$Circumradius=\frac{1}{2}\left(hypotenuse\right)$

$=\frac{1}{2}\times AC=\frac{\sqrt{208}}{6}=\frac{4\sqrt{13}}{6}$

$r=\frac{2\sqrt{13}}{3}$