The Tangent-Radius Perpendicularity Theorem
Simple Explanation
A tangent line touches a circle at exactly one point. At that point of tangency, the tangent line is always exactly perpendicular (at a right angle) to the radius drawn to that same point.
Why Do We Need It?
This is the single most useful fact about tangents — it instantly creates a right angle wherever a tangent meets a radius, unlocking the Pythagorean theorem and trigonometry for tangent-related problems.
See It
A circle with centre O, a radius to point T, and a tangent line through T meeting the radius at a right angle
Formula
Tangent-Radius Perpendicularity
OT ⊥ the tangent line, at the point of tangency T
A tangent line to a circle is always exactly perpendicular to the radius drawn to the point where it touches the circle.
- O
- — the centre of the circle
- T
- — the point of tangency — where the tangent line touches the circle
When to use it: Whenever a tangent line and a radius meet at the point of tangency, guaranteeing a 90° angle there.
Worked Example
Use tangent-radius perpendicularity to find a length
A circle has centre O and radius 6. A tangent from external point P touches the circle at T, with OP = 10. Find PT.
Why Does This Work?
Stated simply: suppose the tangent line met the radius OT at some angle other than 90°. Then the shortest distance from O to the tangent line would be shorter than OT — but the tangent line only touches the circle at the single point T, and every OTHER point on the tangent line must be OUTSIDE the circle (farther from O than the radius). The only way OT can be the shortest possible distance from O to the line is if OT meets the line at exactly a right angle.
Real-Life Example
Designing a satellite dish support strut
An engineer needs a support strut that touches a circular dish at exactly one point, meeting it as cleanly (perpendicular to the dish's radius) as possible.
Tangent-radius perpendicularity is exactly the geometric principle behind designing such a clean, single-point-of-contact strut.
Practice
A circle has radius 5. A tangent from external point P touches the circle at T, with PT = 12. Find OP.
MediumCommon mistake
Assuming any line touching a circle at one point is automatically a tangent — a line could cross through a curved section at a single point without being perpendicular to the radius there; the perpendicularity is what specifically defines a true tangent.
Quick Review
- A tangent is always perpendicular to the radius at the point of tangency.
- This right angle unlocks the Pythagorean theorem for tangent-length problems.
- Proof idea: any other angle would place points of the tangent line inside the circle, contradicting "touches at one point only."