Circles — Tangents and Their Properties
A clear guide to tangents to a circle — the point of contact, the two key theorems, and the tangent-length formula — with a real-life analogy, a worked Pythagoras example and a self-check quiz.
By the PadhoDost Team · 📖 6 min read · Updated 4 August 2026
Part of Class 10 (CBSE) prep🧠 A Cycle Wheel on the Road
Watch a bicycle wheel rolling along a flat road. At any instant the wheel touches the road at just one single point — the road is a tangent to the wheel. And if you draw the spoke from the centre of the wheel to that touching point, it stands perfectly upright, at a right angle to the road. That everyday picture captures the two big ideas of this chapter: a tangent touches a circle at exactly one point, and the radius to that point is perpendicular to the tangent.
Tangent, Secant and Point of Contact
A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact. Compare this with a secant, which is a line that cuts the circle at two points. How many tangents you can draw depends on where your point sits relative to the circle.
| Position of the point | Number of tangents |
|---|---|
| Inside the circle | 0 (no tangent possible) |
| On the circle | 1 (exactly one tangent) |
| Outside the circle | 2 (exactly two tangents) |
The Two Key Theorems
📝 Worked example: length of a tangent
A tangent is drawn from a point P to a circle with centre O and radius 5 cm. The distance OP = 13 cm. Find the length of the tangent PT, where T is the point of contact.
By Theorem 1, the radius OT is perpendicular to the tangent PT, so triangle OTP is right-angled at T.
Apply Pythagoras' theorem: OP^2 = OT^2 + PT^2
13^2 = 5^2 + PT^2
169 = 25 + PT^2, so PT^2 = 169 - 25 = 144
PT = square root of 144 = 12 cm
So the length of the tangent from P is 12 cm.
Remember these
- ✓A tangent touches a circle at exactly one point; a secant cuts it at two points.
- ✓From an external point you can draw exactly two tangents; from a point on the circle, one; from inside, none.
- ✓The radius at the point of contact is perpendicular (90 degrees) to the tangent.
- ✓Tangents drawn from the same external point are equal in length.
- ✓Tangent length = square root of (d^2 - r^2), from Pythagoras' theorem.
⚡ Quick check
From a point P at a distance of 13 cm from the centre of a circle of radius 12 cm, a tangent PQ is drawn. What is the length of PQ?
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