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Circle tangents: the radius is perpendicular

2026-10-04Plane GeometryMs Ho

A tangent meets a circle at one point, and the radius to that point is perpendicular to the tangent.

Key formulas

tangent ⟂ radius at the point of contact
distance from centre to tangent = radius

Core fact

A tangent touches a circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent. That right angle unlocks most tangent calculations.

Worked example

A circle has centre O(0, 0) and radius 5. A tangent touches the circle at T(3, 4). The radius OT has gradient 4/3, so the tangent has gradient −3/4. Using point-slope form at T: y − 4 = (−3/4)(x − 3). Multiplying by 4 gives 4y − 16 = −3x + 9, so 3x + 4y − 25 = 0.

Distance test

A line is tangent to the circle when the perpendicular distance from the centre to the line equals the radius. For 3x + 4y − 25 = 0 and centre (0, 0): |−25| / 5 = 5, which equals the radius, confirming tangency.

Worked example

Worked example

Problem. A circle has centre O(0,0) and radius 5. Show that the line x = 5 is a tangent, and state the point of contact.

Solution. Distance from O to the line x = 5 is 5, equal to the radius, so the line is tangent. The foot of the perpendicular from O is (5,0). Point of contact: (5,0).

Coordinate check

Centre (1, 2), radius 3. A line is tangent iff its distance from (1, 2) is exactly 3.

Review quiz

Circle centre (0,0), radius 4. Which line is tangent to the circle?

Practice questions

  1. Explain why a tangent cannot intersect a circle at two distinct points.
  2. Find the gradient of the tangent at (5, 0) on x² + y² = 25.

Study notes

Is the tangent always unique at a point?

Yes: one tangent line at each point of a smooth circle.

What if distance < radius?

The line cuts the circle in two points (a secant).

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