Tangents and common tangents to circles

Learning outcome

Calculate tangent lengths and angles, distinguishing direct and transverse common tangents.

Concepts and assumptions

A tangent meets a circle at exactly one point and is perpendicular to the radius there. Conversely, a perpendicular to a radius at its circle endpoint is tangent.

For P outside a circle of centre O and radius r, tangents PA and PB touch at A and B. Right triangles OAP and OBP share hypotenuse OP and have equal radii, so:

PA = PB = √(OP² − r²).

Guaranteed equality uses the same external point and circle. Also, ∠APB + minor ∠AOB = 180°.

If ray AT is tangent at A, AB is a chord, and circle point C is opposite T across line AB, the tangent–chord theorem gives ∠TAB = ∠ACB.

For common tangents, take radii R ≥ r > 0 and centre distance d > R + r: circles neither touch nor overlap. Four common tangent lines exist; lengths below mean contact-to-contact segments:

Direct: Both centres lie on the same side of the tangent line. Length = √(d² − (R − r)²).

Transverse: Centres lie on opposite sides; the contact segment crosses the segment joining the centres. Length = √(d² − (R + r)²).

Pythagoras uses hypotenuse d and perpendicular offsets R − r or R + r. Other configurations need separate existence checks.

Worked examples

Example 1 — Two tangent segments. From P, tangents PA and PB touch a circle of centre O and radius 7 cm; OP = 25 cm. Triangle OAP is right-angled at A: PA² = 25² − 7² = 576. Thus PA = PB = 24 cm.

Example 2 — Angle between tangents. From P, tangents PA and PB touch a circle of centre O at A and B, with ∠APB = 64°. Quadrilateral OAPB has right angles at A and B. Thus minor ∠AOB = 360° − 90° − 90° − 64° = 116°.

Example 3 — Direct versus transverse. Two circles have radii 17 cm and 7 cm, with centres 26 cm apart. Since 26 > 17 + 7 = 24, both types exist. Direct length = √(26² − 10²) = √576 = 24 cm. Transverse length = √(26² − 24²) = √100 = 10 cm.

Common mistakes

Use the contact radius for perpendicularity. Do not assume equality across different points or circles. Direct formulas use the radii’s difference; transverse formulas use their sum. Always check separation.

Practice questions

1. P is external to a circle with centre O, radius 9 cm and OP = 15 cm. Find its tangent length.
2. Ray AT is tangent at A; AB is a chord. Point C lies on the circle opposite T across line AB. If ∠TAB = 38°, find ∠ACB.
3. Two externally separated circles have radii 18 cm and 3 cm, with centres 25 cm apart. Find the direct common-tangent length.
4. Two externally separated circles have radii 9 cm and 6 cm, with centres 17 cm apart. Find the transverse common-tangent length.

Worked solutions

1. The contact radius and tangent form a right angle. Tangent² = 15² − 9² = 144, so the length is 12 cm.
2. C is in the required opposite segment. The tangent–chord theorem gives ∠ACB = ∠TAB = 38°.
3. Since 25 > 18 + 3 = 21, separation holds. Direct length = √(25² − (18 − 3)²) = √(625 − 225) = 20 cm.
4. Since 17 > 9 + 6 = 15, transverse tangents exist. Length = √(17² − (9 + 6)²) = √(289 − 225) = 8 cm.