Tangent to a Circle: Definition, Properties & Examples

A tangent to a circle is a straight line that touches the circle at exactly one point. Imagine a bicycle tyre rolling along the road. At every single instant, the road touches the tyre at just one point, not two, not zero, exactly one. This contact point changes every split second as the wheel rolls forward, but the relationship stays the same. This is the geometry of a tangent to a circle in real life. In this guide, you will learn about the tangent to a circle, its definition, and properties through solved examples and practice questions.

Table of Contents

• What is the Tangent to a Circle
• Number of Tangents to a Circle from a Point
• Properties of Tangent to a Circle
• Solved Examples on Tangent to a Circle

What is the Tangent to a Circle

A tangent to a circle is a line that intersects the circle at only one point. The tangent touches the circle at the common point of the tangent and the circle, which is called the 'point of contact'.

Number of Tangents to a Circle from a Point

The number of distinct tangents that can be drawn from a given external point to a circle depends on its position relative to the circle.

Point inside the circle: Every line through an interior point cuts across the circle, entering through one side and exiting through another. Such lines form a secant. No tangent can be drawn from here.

∴ 0 tangents are possible from a point inside the circle. There is no tangent to a circle passing through a point lying inside the circle.

Point on the circle: The point is already sitting on the circumference. You can draw exactly one tangent at this point. The line just grazes the circle at that location without going inside. There is one and only one tangent to a circle passing through a point lying on the circle.

Point outside the circle: From a point beyond the circle, you can draw two distinct tangents. They reach the circle from either side, touching it at two different points of contact. There are exactly two tangents to a circle through a point lying outside the circle.

Properties of Tangent to a Circle

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

If two tangents are drawn from a point outside the circle, their lengths are equal. If P is an external point, and PA and PB are tangents touching the circle at A and B, then PA = PB always.

A circle can have a maximum of two parallel tangents, and they are always drawn at the two ends of a diameter. For two tangents to be parallel, their corresponding radii must both be perpendicular to the same line, meaning the radii must lie along the same direction. This occurs only when the radii form a single diameter, placing the points of contact at opposite ends of that diameter.

Solved Examples on Tangent to a Circle

Example 1: Tangent PQ at point P on a circle (r = 5 cm) meets a line through centre O at Q where OQ = 12 cm. Find PQ.

Solution: PQ is tangent at P. ∴ OP ⊥ PQ and ∠OPQ = 90° (∵ the tangent at any point of a circle is perpendicular to the radius)

In right-angled △OPQ, OQ² = OP² + PQ²

12² = 5² + PQ²

144 = 25 + PQ²

PQ² = 119

PQ = √119 cm

∴ Length of tangent PQ = √119 cm ≈ 10.9 cm

Example 2: From point Q, the length of the tangent to a circle is 24 cm, and the distance of Q from the centre is 25 cm. Find the radius.

Solution: Let tangent touch at P. ∴ OP ⊥ QP and ∠OPQ = 90° (∵ the tangent at any point of a circle is perpendicular to the radius)

OQ² = OP² + QP²

25² = r² + 24²

625 = r² + 576

r² = 49

r = 7 cm

∴ radius = 7 cm

Example 3: TP and TQ are tangents drawn to a circle with centre O from external point T. Prove that ∠PTQ = 2∠OPQ.

Solution: Let ∠PTQ = θ. Since TP = TQ (∵ two tangents from an external point are equal), ∴ △TPQ is isosceles.

∠TPQ = ∠TQP = (180° − θ) / 2 = 90° − θ/2

∠OPT = 90° (∵ the tangent at any point of a circle is perpendicular to the radius)

∠OPQ = ∠OPT − ∠TPQ = 90° − (90° − θ/2) = θ/2

Therefore, ∠PTQ = θ = 2 × (θ/2) = 2∠OPQ

Example 4: Two concentric circles have radii 5 cm and 3 cm. A chord AB of the larger circle is tangent to the smaller circle at C. Find the length of AB.

Solution: AB is tangent to the smaller circle at C, so OC ⊥ AB (∵ the tangent at any point of a circle is perpendicular to the radius), where O is the common centre.

Since OC ⊥ chord AB of the larger circle, OC bisects AB. So AC = CB = ½ AB.

In right-angled △OCA: OA² = OC² + AC²

5² = 3² + AC²

25 − 9 = AC²

AC² = 16

AC = 4 cm

AB = 2 × AC = 2 × 4 = 8 cm

∴ length of the chord = 8 cm