Tangent Perpendicular to Radius

Statement

The tangent to a circle at any point is perpendicular to the radius drawn to that point of contact. In other words, if a line touches a circle at exactly one point, then the radius to that point makes a right angle (90°) with the tangent line.

\[ \text{Radius} \perp \text{Tangent at the point of contact} \]

Why it’s true

• A tangent is a line that touches the circle at exactly one point.
• The shortest line from the circle’s centre to this tangent is the radius, and shortest lines meet at right angles.
• This property is fundamental and is often used to connect circle theorems with right-angled triangles and Pythagoras’ theorem.

Recipe (how to use it)

1. Locate the tangent and the point of contact with the circle.
2. Draw the radius to that point.
3. Mark the angle between radius and tangent as 90°.
4. Use this right angle in solving triangle problems, applying Pythagoras or trigonometry if required.

Spotting it

Whenever you see a tangent touching a circle, immediately think: “That’s perpendicular to the radius.” The 90° symbol is a strong visual clue in exam diagrams.

Common pairings

• Right-angled triangles with Pythagoras’ theorem.
• Trigonometry in right-angled triangles.
• Other circle theorems (alternate segment theorem, angles in semicircle, etc.).

Mini examples

1. Given: A circle radius 5 cm, tangent touches at point A. What is ∠OAP (where O is centre, P is a point on tangent)? Answer: 90°.
2. Given: A right triangle formed by radius 4 cm, tangent segment 3 cm. Find: hypotenuse. Answer: 5 cm (Pythagoras).

Pitfalls

• Forgetting that “tangent” means exactly one point of contact.
• Not marking the 90° symbol in working diagrams.
• Confusing tangents with secants (which cut the circle twice).

Exam strategy

• Look for tangents in diagrams — it often signals a right-angled triangle hidden in the problem.
• Always mark 90° to unlock Pythagoras or trigonometric ratios.

Summary

The tangent to a circle is perpendicular to the radius at the point of contact. This simple but powerful theorem is a gateway to solving many circle geometry problems, especially those involving right triangles, distances, and angle proofs.