Concept 10.2
The Tangent Is Perpendicular to the Radius
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
Theorem 10.1: the tangent at any point of a circle is perpendicular to the radius through the point of contact.
The intuition
Of all the points on the tangent, the point of contact is the one closest to the centre: every other point on the tangent lies outside the circle, so it's farther away.
The shortest segment from a point to a line is the perpendicular one. So the radius to the point of contact meets the tangent at 90°.
The attention
Look at the radius that goes to the point of contact.
OP ⊥ tangent at P not OQ, to some other point
Look hereNot here
- Look here
- The radius to the point of contact. Only that radius is perpendicular to the tangent. It is where the right angle you need is hiding.
- Not here
- Somewhere else. Other radii meet the tangent at other angles, or not at all. The 90° is special to the point of contact.
- What it teaches
- Before you look for a right angle, find the point where it must be.