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Lesson 2 of 4

Circles · Lesson 2 of 4

Tangent to a Circle

“A tangent is geometry's ultimate smooth operator—it touches the circle at exactly one point and leaves without any awkward drama.”

Learning Objectives

• State Theorem 10.1 correctly. • Explain its proof step by step. • Use the tangent–radius perpendicularity result. • Recognize the normal at the point of contact. • Solve tangent-length questions using Pythagoras theorem.

Tangent

A tangent is a straight line that touches a circle at exactly one point without cutting through it. That point is called the point of contact.

You can imagine placing a ruler so that it just touches the edge of a circular coin. If the ruler touches the coin at only one point, it acts like a tangent. If the ruler passes through the circle and meets it at two points, then it is not a tangent.

Tangents have a very important relationship with the radius of the circle. If we draw a radius from the centre of the circle to the point where the tangent touches it, the radius and the tangent always meet at a right angle.

So, if a tangent touches a circle at point P and OP is the radius drawn to that point, then:

OP ⟂ tangent at P

This simple property is extremely useful. It allows us to identify tangents, construct right angles, and solve many geometry problems involving circles.

In this lesson, we will explore why the tangent is perpendicular to the radius at the point of contact and how this property can be used in different situations.

Theorem

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

If XY is tangent at P to a circle with centre OLaTeX
Tangent Perpendicular to Radius Theorem 10.1: OP ⊥ XY at point of contact P O P X Y Radius (r) OP ⊥ XY
Tangent perpendicular to radius— centre O, tangent XY touching at P, radius OP and a right-angle marker at P.

Proof of Theorem

Let XY be tangent to a circle with centre O at P. We must prove OP ⟂ XY.

Take any other point Q on XY and join OQ. Q must lie outside the circle; otherwise XY would cut the circle at two points and become a secant.

Since Q lies outside the circle, OQ is longer than the radius OP. Thus OQ > OP for every point Q on XY other than P.

So OP is the shortest distance from O to line XY. The shortest distance from a point to a line is perpendicular. Therefore OP ⟂ XY.

Shortest-Distance Proof (Theorem 10.1) OP is the shortest distance from O to XY, so OP ⊥ XY O P Q X Y OP (Radius) OQ > OP Since Q is outside circle, OQ > OP ⇒ OP ⊥ XY
Shortest-distance proof
Remarks

• At any point on a circle there is one and only one tangent. • The line containing the radius through the point of contact is also called the normal.

Worked Example: Find the Tangent Length

Problem
PQ is tangent at P to a circle of radius 5 cm. OQ = 13 cm. Find PQ.

  1. 1.OP ⟂ PQ by Theorem 10.1.
  2. 2.So △OPQ is right-angled at P.
  3. 3.OP = 5 cm, OQ = 13 cm.
  4. 4.OQ² = OP² + PQ².
  5. 5.13² = 5² + PQ².
  6. 6.169 = 25 + PQ².
  7. 7.PQ² = 144.
  8. 8.PQ = 12 cm.
Right Triangle Formed by a Tangent ΔOPQ is right-angled at P (∠OPQ = 90°) O P Q Radius (OP) Tangent (PQ) Hypotenuse (OQ) Pythagoras Theorem OQ² = OP² + PQ²
Right triangle formed by a tangent— OQ as hypotenuse, OP as radius, PQ as tangent and a right angle at P.
Worked Example: Find the Radius

Problem
A tangent AP has length 12 cm and AO = 13 cm. Find the radius.

  1. 1.OP ⟂ AP.
  2. 2.So △AOP is right-angled at P.
  3. 3.AO² = AP² + OP².
  4. 4.13² = 12² + r².
  5. 5.169 = 144 + r².
  6. 6.r² = 25.
  7. 7.r = 5 cm.
Worked Example: Tangents at the Ends of a Diameter

Problem
Prove that tangents drawn at the endpoints of a diameter are parallel.

  1. 1.Let AB be a diameter.
  2. 2.The tangent at A is perpendicular to OA.
  3. 3.The tangent at B is perpendicular to OB.
  4. 4.OA and OB lie on the same straight line AB.
  5. 5.Both tangents are perpendicular to the same line, so they are parallel.
Tangents at the Endpoints of a Diameter Tangents PQ and RS drawn at endpoints A and B are parallel (PQ ∥ RS) O A B P Q R S Diameter AB ∠PAB = ∠ABS = 90° (Alternate Angles ⇒ PQ ∥ RS)
Tangents at the endpoints of a diameter— diameter AB through centre O and tangents at A and B with right-angle markers.

Quiz

Quick check

A line touches a circle at exactly one point. What is this line called?

Quick check

What is the point at which a tangent touches a circle called?

Quick check

A tangent touches a circle at T, and O is the centre of the circle. What is the value of ∠OTP?

Quick check

How many tangents can be drawn to a circle from a point lying on the circle?

Quick check

A tangent PT is drawn from an external point P to a circle with centre O. If OP = 17 cm and the radius OT = 8 cm, what is the length of PT?

Practice Problems

Practice Problems
  1. A tangent PQ at P to a circle of radius 7 cm has OQ = 25 cm. Find PQ.
  2. Prove that tangents at the endpoints of a diameter are parallel.
  3. Prove that the perpendicular at the point of contact to a tangent passes through the centre.
  4. A tangent from A is 15 cm long and AO = 17 cm. Find the radius.

Key Takeaways

Key Takeaways

• A tangent is perpendicular to the radius at the point of contact. • This creates a right triangle when the centre is joined to an external point. • Pythagoras theorem is a key tool in tangent-length problems. • The radius through the point of contact is also called the normal.

Previous · Lesson 1

Circles

Next · Lesson 3

Number of Tangents from a Point on a Circle