Question

In a circle with centre O, AB and CD are two diameters perpendicular to each other. The length of chord AC is:

• A. 2AB
• B. √2 AB
• C. 1/2AB
• D. (1/√2)AB

Answer 1

The correct option is

D

(1/√2)AB

Given: In a circle with centre O, AB and CD are two diameters perpendicular to each other

$A{C}^2=O{A}^2+O{C}^2=2O{A}^2$.....(i)
Since $OC=OA=\frac{AB}{2}$
squaring and multiplying 2 on both sides, we get
$2O{A}^2=2(\frac{AB}{2}{)}^2=\frac{A{B}^2}{2}$
Substitute $2O{A}^2=\frac{A{B}^2}{2}$ in equation(i), we get
$A{C}^2=\frac{A{B}^2}{2}$
$\therefore AC=\frac{1}{\sqrt{2}}AB$