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Lesson 3 of 4

Circles · Lesson 3 of 4

Number of Tangents from a Point on a Circle

“Inside the circle: Good luck drawing a tangent—it’s a total zero-contact zone in here!”

Learning Objectives

• State the number of tangents possible from points inside, on and outside a circle. • Define tangent length. • State and prove Theorem: Lengths of tangents from an external point are equal. • Use equal tangents in length and angle problems. • Understand properties of concentric circles and intersecting tangents.

Tangents from a Point on a Circle

The number of tangents that can be drawn from a point to a circle depends on where that point lies.

Imagine choosing a point somewhere near a circle and trying to draw straight lines from that point that just touch the circle without cutting through it. The result changes depending on the position of the point.

If the point lies inside the circle, no tangent can be drawn because every line through the point will pass through the circle. If the point lies on the circle, exactly one tangent can be drawn at that point. If the point lies outside the circle, two different tangents can be drawn, each touching the circle at a different point.

So, there are three important cases to study:

Point inside the circle → 0 tangents

Point on the circle → 1 tangent

Point outside the circle → 2 tangents

Understanding these three situations helps us see how the position of a point affects its relationship with a circle and prepares us for an important result: the two tangents drawn from the same external point have equal lengths.

Position of pointNumber of tangents
Inside the circle0
On the circle1
Outside the circle2
Number of Tangents from Different Point Positions Inside the circle: 0 tangents • On the circle: 1 tangent • Outside the circle: 2 tangents Case 1: Point Inside No tangent can be drawn O P 0 tangents Case 2: Point on the Circle Exactly one tangent O Q radius 1 tangent Case 3: Point Outside Exactly two tangents O R A B 2 tangents
Number of tangents from different point positions— Three cases: point inside with no tangent, point on circle with one tangent, and external point with two tangents.
Definition
Length of Tangent

The length of the segment joining an external point to the point of contact on the circle.

Theorem: Equal Tangents from an External Point

Theorem

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

If PQ and PR are tangents from PLaTeX
Two Equal Tangents from an External Point Theorem 10.2: Tangent lengths PQ = PR O P Q R PQ = PR and ΔOQP ≅ ΔORP
Two equal tangents from an external point— Centre O, external point P, contact points Q and R, tangents PQ and PR, radii OQ and OR, and right-angle markers.

Proof of Theorem

Join OP, OQ and OR. Since Q and R are points of contact, OQ ⟂ PQ and OR ⟂ PR by Theorem 10.1.

Thus △OQP and △ORP are right triangles. Also, OQ = OR because they are radii of the same circle, and OP is common.

Therefore △OQP ≅ △ORP by RHS congruence. Hence PQ = PR by corresponding parts of congruent triangles.

Useful Consequences

• The theorem can also be proved using Pythagoras theorem. • OP bisects the angle between the two tangents because ∠OPQ = ∠OPR.

Example: Concentric Circles

Chord Touching the Smaller Circle

Problem
Two concentric circles have centre O. Chord AB of the larger circle touches the smaller circle at P. Prove AP = PB.

  1. 1.AB is tangent to the smaller circle at P.
  2. 2.OP is the radius through the point of contact.
  3. 3.By Theorem 10.1, OP ⟂ AB.
  4. 4.AB is a chord of the larger circle.
  5. 5.A perpendicular from the centre to a chord bisects the chord.
  6. 6.Therefore AP = PB.
Chord of Outer Circle Tangent to Inner Circle OP ⊥ AB ⇒ Point of contact P bisects chord AB (AP = PB) O P A B C₁ (Outer) C₂ (Inner) OP ⊥ AB (Theorem 10.1) ∴ AP = PB (Chord Bisected)
Chord of larger concentric circle tangent to smaller circle— Two concentric circles, chord AB of outer circle tangent to inner circle at P, and OP perpendicular to AB.

Example: Angle Between Two Tangents

Prove an Angle Relation

Problem
Tangents TP and TQ are drawn from external point T. Prove ∠PTQ = 2∠OPQ.

  1. 1.Let ∠PTQ = θ.
  2. 2.TP = TQ by Theorem 10.2, so △TPQ is isosceles.
  3. 3.Hence ∠TPQ = ∠TQP = (180° − θ)/2 = 90° − θ/2.
  4. 4.By Theorem 10.1, ∠OPT = 90°.
  5. 5.So ∠OPQ = ∠OPT − ∠TPQ = 90° − (90° − θ/2) = θ/2.
  6. 6.Therefore θ = 2∠OPQ, so ∠PTQ = 2∠OPQ.
Angle Relation Between Tangents & Chord Angle between tangents is twice the angle made by chord with radius O T P Q ∠PTQ ∠OPQ ∠PTQ = 2 ∠OPQ
Two tangents and angle relation— External point T, tangents TP and TQ, center O and radii OP and OQ. Mark ∠PTQ and ∠OPQ.

Example: Tangents from the Ends of a Chord

Find TP

Problem
PQ is a chord of length 8 cm in a circle of radius 5 cm. Tangents at P and Q meet at T. Find TP.

  1. 1.Join OT and let it meet PQ at R.
  2. 2.TP = TQ, so △TPQ is isosceles.
  3. 3.OT bisects ∠PTQ and is perpendicular to PQ, so R is the midpoint of PQ.
  4. 4.PR = RQ = 4 cm.
  5. 5.In right △OPR, OR = √(5² − 4²) = 3 cm.
  6. 6.Right triangles TRP and PRO are similar.
  7. 7.Therefore TP/PO = RP/RO.
  8. 8.TP/5 = 4/3.
  9. 9.TP = 20/3 cm.
Angle Relation Between Tangents & Chord Angle between tangents is twice the angle made by chord with radius O T P Q ∠PTQ ∠OPQ ∠PTQ = 2 ∠OPQ
Chord with tangents meeting at an external point— Chord PQ, tangents at P and Q meeting at T, centre O, OT meeting PQ at R, and right angle at R.
Circumscribed Quadrilateral

Problem
A quadrilateral ABCD circumscribes a circle. Prove AB + CD = AD + BC.

  1. 1.Let contact points on AB, BC, CD, DA be P, Q, R, S.
  2. 2.AP = AS, BP = BQ, CQ = CR, DR = DS.
  3. 3.AB + CD = (AP + PB) + (CR + RD).
  4. 4.Replacing equal tangent segments gives (AS + BQ) + (CQ + DS).
  5. 5.Rearranging gives (AS + SD) + (BQ + QC) = AD + BC.
  6. 6.Hence AB + CD = AD + BC.
Quadrilateral Circumscribing a Circle Prove: AB + CD = AD + BC O P Q R S A B C D Proof 1. Let the circle touch AB, BC, CD, DA at P, Q, R, S. 2. Tangents from an external point are equal in length: AP = AS, BP = BQ, CQ = CR, DR = DS 3. AB + CD = (AP + PB) + (CR + RD) 4. Replace each segment with its equal tangent length: = (AS + BQ) + (CQ + DS) 5. Regroup the four terms around D and B: = (AS + SD) + (BQ + QC) 6. = AD + BC Therefore, AB + CD = AD + BC Tick marks in the figure show each equal tangent-segment pair (1, 2, 3, 4 ticks), and dashed radii meet each side at 90°.
Circumscribed quadrilateral— ABCD around a circle with contact points P, Q, R, S and labels outside the sides.
Circumscribed Triangle

Problem
A triangle ABC circumscribes a circle. D on BC gives BD = 8 cm and DC = 6 cm. If the tangent length from A is 4 cm, find AB and AC.

  1. 1.Let contact points on AB and AC be F and E.
  2. 2.BF = BD = 8 cm.
  3. 3.CE = CD = 6 cm.
  4. 4.AF = AE = 4 cm.
  5. 5.AB = 4 + 8 = 12 cm.
  6. 6.AC = 4 + 6 = 10 cm.
Triangle Circumscribing a Circle (Incircle) Equal tangent pairs: AF = AE = x, BD = BF = y, CD = CE = z O A B C D E F x x y y z z AF = AE = x , BD = BF = y , CD = CE = z
Triangle circumscribing a circle— Triangle ABC around a circle touching BC at D, CA at E and AB at F. Indicate equal tangent pairs.
Common Mistakes

• Equal tangent lengths apply only to tangents from the same external point. • Radius and tangent are perpendicular only at the point of contact. • In angle questions, remember the two right angles at contact points. • In circumscribed polygons, pair tangent segments vertex by vertex.

Quiz

Quick check

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

Quick check

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

Quick check

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

Quick check

A circle has centre O and radius 6 cm. A point P is 4 cm from O. How many tangents can be drawn from P to the circle?

Quick check

A circle has centre O and radius r. Which statement about a point P is correct?

Practice Problems

Practice Problems
  1. From Q, a tangent to a circle is 24 cm and Q is 25 cm from the centre. Find the radius.
  2. If ∠POQ = 110° for two tangents TP and TQ, find ∠PTQ.
  3. If tangents PA and PB make an angle of 80°, find ∠POA.
  4. Two concentric circles have radii 5 cm and 3 cm. Find the chord length of the larger circle touching the smaller.
  5. Prove AB + CD = AD + BC for a circumscribed quadrilateral.
  6. Prove that the angle between two tangents is supplementary to the central angle subtended by the segment joining the contact points.
  7. Prove that a parallelogram circumscribing a circle is a rhombus.
  8. A triangle ABC circumscribes a circle. D divides BC into 8 cm and 6 cm. Find AB and AC when the tangent length from A is 4 cm.

Key Takeaways

Key Takeaways

• A point inside gives 0 tangents, on the circle gives 1, and outside gives 2. • Tangents from the same external point are equal. • The centre lies on the angle bisector of the angle between the tangents. • Equal tangent segments are powerful in circumscribed figures.

Previous · Lesson 2

Tangent to a Circle

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Chapter Summary and Practice