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Lesson 4 of 4

Circles · Lesson 4 of 4

Chapter Summary and Practice

“Tangents, radii and their elegant relationships return for one final trip around the circle.”

Learning Objectives

• By the end of this lesson, you should be able to. • Recall the main tangent definitions and theorems. • Choose the correct tangent property for a problem. • Solve mixed length, angle and proof questions. • Use equal tangent segments in circumscribed figures.

This chapter is built around three ideas: what a tangent is, why it is perpendicular to the radius at the point of contact, and why tangents from the same external point are equal.

Tangent, Secant and Point of Contact

A secant cuts a circle at two points. A tangent touches the circle at exactly one point called the point of contact.

Theorem

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

Tangent–radius relationLaTeX

Number of Tangents

Point liesTangents
Inside circle0
On circle1
Outside circle2

Theorem

The lengths of tangents drawn from the same external point to a circle are equal.

Equal tangent lengthsLaTeX

How to Choose the Right Method

What you noticeUse
Radius meets tangent at contact pointTheorem 10.1
Centre, tangent and external point form a right trianglePythagoras theorem
Two tangents from one external pointTheorem 10.2
Angle between two tangentsTwo right angles + angle sum
Circumscribed polygonEqual tangent segments from each vertex
Chord tangent to inner concentric circlePerpendicular from centre bisects chord
Common Mistakes

• Equal tangents must start from the same external point. • Do not forget the right angle between radius and tangent. • Do not confuse a tangent with a secant. • In circumscribed figures, pair equal tangent segments vertex by vertex.

Guided Practice

Tangent Length

Problem
A point P is 15 cm from the centre of a circle of radius 9 cm. Find the tangent length.

  1. 1.Let PT be tangent and OT the radius.
  2. 2.OT ⟂ PT.
  3. 3.PT² = OP² − OT² = 15² − 9² = 144.
  4. 4.PT = 12 cm.
Angle Between Tangents

Problem
Tangents PA and PB are drawn from P. If ∠AOB = 100°, find ∠APB.

  1. 1.∠OAP = ∠OBP = 90°.
  2. 2.100° + 90° + 90° + ∠APB = 360°.
  3. 3.∠APB = 80°.
Concentric Circles

Problem
Two concentric circles have radii 17 cm and 8 cm. A chord of the larger circle touches the smaller. Find the chord length.

  1. 1.Let OP be perpendicular to the chord at P.
  2. 2.The perpendicular from the centre bisects the chord.
  3. 3.Let half the chord = x.
  4. 4.17² = 8² + x².
  5. 5.x² = 225, so x = 15.
  6. 6.Chord length = 30 cm.
Circumscribed Quadrilateral

Problem
ABCD circumscribes a circle. AB = 12 cm, BC = 9 cm, CD = 7 cm. Find AD.

  1. 1.AB + CD = AD + BC.
  2. 2.12 + 7 = AD + 9.
  3. 3.AD = 10 cm.
Circumscribed Triangle

Problem
D on BC gives BD = 8 cm and DC = 6 cm. Tangent length from A is 5 cm. Find AB and AC.

  1. 1.Let contact points on AB and AC be F and E.
  2. 2.AF = AE = 5 cm.
  3. 3.BF = BD = 8 cm.
  4. 4.CE = CD = 6 cm.
  5. 5.AB = 13 cm and AC = 11 cm.

Quiz

Quick check

A tangent has how many common points with a circle?

Quick check

The radius to the point of contact is:

Quick check

How many tangents can be drawn from a point outside a circle?

Quick check

If PQ and PR are tangents from P, then:

Quick check

If the central angle between contact points is 120°, the angle between the tangents is:

Before You Finish the Chapter

Make sure you can distinguish tangent and secant, use the tangent–radius right angle, find tangent length with Pythagoras theorem, use equal tangent segments, and solve angle questions.

Practice Problems

Practice Questions
  1. How many tangents can a circle have?
  2. A tangent at P to a circle of radius 8 cm has OQ = 17 cm. Find PQ.
  3. Draw one tangent and one secant parallel to each other.
  4. Prove that tangents at the endpoints of a diameter are parallel.
  5. A point is 13 cm from the centre and its tangent length is 12 cm. Find the radius.
  6. Two concentric circles have radii 10 cm and 6 cm. Find the chord length of the larger circle touching the smaller.
  7. If ∠POQ = 108° for two tangents TP and TQ, find ∠PTQ.
  8. If tangents PA and PB make an angle of 82°, find ∠POA.
  9. Prove AB + CD = AD + BC for a circumscribed quadrilateral.
  10. Prove that the angle between two tangents is supplementary to the central angle subtended by the segment joining contact points.
  11. Prove that a parallelogram circumscribing a circle is a rhombus.
  12. A triangle circumscribes a circle and D divides BC into 9 cm and 7 cm. If the tangent length from A is 5 cm, find AB and AC.

Key Takeaways

Key Takeaways

• A tangent touches a circle at exactly one point. • The tangent is perpendicular to the radius through the point of contact. • From an external point, exactly two tangents can be drawn. • Those tangent lengths are equal. • Most problems reduce to right triangles, congruence, angle sums or equal tangent segments.

Previous · Lesson 3

Number of Tangents from a Point on a Circle

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