Radius–Tangent Theorem
A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency or point of contact. The Radius–Tangent Theorem is one of the most fundamental results in Euclidean circle geometry: it states that the radius drawn to the point of tangency is always perpendicular to the tangent line. In other words, the angle between the radius and the tangent at the point of contact is exactly 90°. This theorem is not just a curiosity — it is a powerful tool used throughout geometry, trigonometry, and even physics and engineering. Whenever you need to find lengths, angles, or relationships involving tangent lines and circles, this theorem almost certainly appears in the solution. It connects the abstract idea of a tangent (a line that barely 'touches' a curve) to a precise, measurable right-angle relationship. Understanding this theorem requires knowing what it means for two lines to be perpendicular and being comfortable with proof by contradiction — a classic technique in deductive reasoning. The proof is elegant: it shows that if the radius were not perpendicular to the tangent, then another point on the tangent line would be closer to the centre than the radius length, forcing that point to lie inside the circle. But a tangent only touches the circle at one point, which gives a contradiction. This style of rigorous argument is the heart of Euclidean geometry.
- Key Definitions
- Statement of the Theorem
- Proof by Contradiction
- Converse of the Theorem
- Practical Consequences
Key Definitions
A circle with centre O and radius r is the set of all points exactly distance r from O. A tangent to the circle at point P is a line ℓ that intersects the circle at P and at no other point. The radius OP is the line segment from the centre O to the point of tangency P.
Statement of the Theorem
Radius–Tangent Theorem: If ℓ is a tangent to a circle with centre O at the point of tangency P, then OP ⊥ ℓ. Equivalently, the angle ∠OPT = 90° for any point T on the tangent line different from P.
Proof by Contradiction
Assume, for contradiction, that OP is NOT perpendicular to ℓ. Then we can drop a perpendicular from O to ℓ, meeting ℓ at some point Q ≠ P. In the right triangle OQP (right angle at Q), the hypotenuse is OP. By the Pythagorean theorem, OQ² + QP² = OP², so OQ² = OP² − QP² < OP² = r². Therefore OQ < r, meaning Q is strictly inside the circle. But Q lies on ℓ. Since Q is inside the circle and P is on the circle, the line ℓ must cross the circle again at a second point (by the Intermediate Value Theorem / continuity). This contradicts ℓ being a tangent (a tangent meets the circle at exactly one point). Therefore our assumption was wrong, and OP must be perpendicular to ℓ.
Converse of the Theorem
The converse is also true and equally useful: if a line ℓ passes through a point P on a circle and is perpendicular to the radius OP at P, then ℓ is a tangent to the circle at P. This is proven similarly — any other point on ℓ is farther from O than r (since OP is the shortest distance from O to ℓ), so no other point on ℓ lies on or inside the circle.
Practical Consequences
Because OP ⊥ ℓ, triangle OPT is a right triangle for any point T on the tangent. This means: (1) You can use Pythagoras' theorem to relate the tangent length, the radius, and the distance from the centre to an external point. Specifically, if T is an external point and PT is a tangent segment, then OT² = OP² + PT², i.e. OT² = r² + PT². (2) Two tangents drawn from the same external point T to a circle have equal length: |TA| = |TB| where A and B are the two points of tangency. This follows because triangles OAT and OBT are both right-angled at A and B respectively, share hypotenuse OT, and have equal legs OA = OB = r.
The Key Equation
- OT — Distance from centre to external point (length units): The straight-line distance from the centre O of the circle to the external point T.
- r — Radius (length units): The radius of the circle; also the length of segment OP where P is the point of tangency.
- PT — Tangent length (length units): The length of the tangent segment from the external point T to the point of tangency P.
Worked Example
Problem. A circle has centre O and radius 5 cm. An external point T is 13 cm from the centre O. A tangent from T touches the circle at point P. Find the length of the tangent segment PT, and state the size of angle OPT.
Answer. PT = 12 cm; angle OPT = 90°
Origins · 1 minEuclid of Alexandria and how this was discovered
Euclid compiled and systematised Greek mathematical knowledge in his thirteen-volume work 'Elements' (c. 300 BCE). Book III of the Elements is devoted entirely to the geometry of circles and contains Proposition 16, which establishes the relationship between a tangent and the diameter at the point of contact — the foundation for the Radius–Tangent Theorem. His axiomatic, deductive approach to geometry set the standard for mathematical proof for over two thousand years. When the Egyptian pharaoh Ptolemy I asked Euclid if there was a shortcut to learning geometry, Euclid reportedly replied: 'There is no royal road to geometry.' This saying is still quoted today to mean that there are no shortcuts to genuine understanding.
Related Concepts
- Chord Properties — Chords and tangents are both fundamental line–circle relationships; the perpendicular from the centre to a chord bisects it, which parallels the perpendicularity result for tangents.
- Tangents from an External Point — Uses the Radius–Tangent Theorem directly to prove that two tangent segments from one external point are equal in length.
- Pythagoras' Theorem — The 90° angle guaranteed by the Radius–Tangent Theorem creates a right triangle, making Pythagoras' Theorem the primary tool for calculating lengths.
- Proof by Contradiction — The standard proof of the Radius–Tangent Theorem is a classic example of proof by contradiction (reductio ad absurdum).
- Angle in a Semicircle — Both theorems produce 90° angles associated with circles and are proved using similar Euclidean deductive techniques.