Concept 10.4
The Right Triangle in Every Tangent Problem
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
Combining the two theorems: if a tangent from P touches a circle with centre O at A, then OA ⊥ PA, so OP² = OA² + PA².
The intuition
Every tangent problem hides a right triangle: the radius, the tangent, and the line to the centre. The right angle is at the point of contact.
With radius 3 cm and OP = 5 cm, the tangent length is √(5² − 3²) = 4 cm. The line to the centre is the hypotenuse, so the radius is subtracted from it, not added.
The attention
Look at the right angle at the point of contact: it makes OP the hypotenuse.
∠OAP = 90° ⇒ PA² = OP² − OA² not OP² + OA²
Look hereNot here
- Look here
- The right angle at the point of contact. The right angle at A makes OP the longest side. Then PA² = OP² − OA².
- Not here
- Adding the squares. Adding the squares treats PA as the hypotenuse. It isn't: PA is a leg.
- What it teaches
- Locate the right angle first; it tells you which side is the hypotenuse.