Question

In a circle with centre O a tangent is drawn from an external point $A$ to the circle which touches the circle at $B$. If $OB=5cm,OA=13cm$, then the length of $AB=$

• A. 6.5 cm
• B. 6 cm
• C. 13 cm
• D. 12 cm

Answer 1

The correct option is D 12 cm


In the circle with centre $O,AB$ is a tangent and $OB$ is a radius passing through point of contact.
$\therefore OB\perp AB,\therefore \angle OBA={90}^{\circ }$
$\therefore △AOB$ is a right-angled triangle, of which $OA$ is the hypotenuse.
$\therefore {OB}^2+{AB}^2={OA}^2$ [by pythagoras' theorem]
or, $5^2+{AB}^2={13}^2[\because OB=5cm,OA=13cm]$
or, $25+{AB}^2=169$ or, ${AB}^2=169-25$
or, $A{B}^2=144$ or, $AB=\sqrt{144}$ or, $AB=12$
Hence, the length of $AB=12cm$