Question

AP is the tangent to the circle with centre O and radius 8 cm . If AB = 9 cm, then find the length of tangent AP.

Answer 1

The tangent at any point of a circle is perpendicular to the radius through the point of contact.
$\therefore AP\perp OP$

$\Rightarrow \angle OPA={90}^{\circ }\text{and}△OPA\text{is a}\text{right-angled triangle.}$


$OA=OB+AB=8+9=17cm$

$\text{Now, applying pythagoras theorem in}\text{triangle}OPA,$
$(OP{)}^2+(AP{)}^2=\left(OA{)}^2\right(AP{)}^2=(OA{)}^2-(OP{)}^2(AP{)}^2=(17{)}^2-\left(8{)}^2\right(AP{)}^2=225\left(AP\right)=\sqrt{225}AP=15cm$