The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
Step-by-Step Solution
Key Concept: From a point outside a circle, the line joining the point to the centre and the tangent drawn from the point form a right‑angled triangle, because the radius drawn to the point of tangency is perpendicular to the tangent. Hence we can apply the Pythagorean theorem.
1. Let O be the centre of the circle and let P be the point of contact of the tangent on the circle.
2. Join O to A and O to P. Then \(OP\) is the radius (say \(r\)) and \(AP\) is the given tangent length = 4 cm.
3. Since a radius drawn to the point of tangency is perpendicular to the tangent, \(\angle OPA = 90^{\circ}\). Thus \(\triangle OPA\) is a right‑angled triangle with hypotenuse \(OA\) = 5 cm.
4. Apply Pythagoras theorem:
$$ OA^{2} = OP^{2} + AP^{2} $$
Substituting the known values:
$$ 5^{2} = r^{2} + 4^{2} $$
$$ 25 = r^{2} + 16 $$
$$ r^{2} = 25 - 16 = 9 $$
5. Hence \(r = \sqrt{9} = 3\) cm.
6. Therefore, the radius of the circle is 3 cm.
Correct Answer: Radius = 3 cm