The length of the tangent from a point A at a distance of 13cm from the centre of the circle is 12cm. What is the radius of the circle?

Let the centre of circle be O so that AO = 13 cm

Tangent is AB whose length is 12 cm

OB is radius as shown

Now we know that radius is perpendicular to point of contact

Hence OB is perpendicular to AB

Hence ∠ABO = 90°

Consider ΔABO

Using Pythagoras theorem

⇒ AB² + OB² = AO²

⇒ 12² + OB² = 13²

⇒ OB² = 13² - 12²

⇒ OB² = 169 – 144

⇒ OB² = 25

⇒ OB = ±5

As length cannot be negative

⇒ OB = 5 cm

Hence length of radius is 5 cm