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Chords and tangents of a circle: Class 10 Mathematics
Circle terms, theorems on chords and their distance from the centre, tangents and the radius, equal tangents from an external point, the tangent chord angle, and circles touching each other.
Key points
A circle is the set of all points in a plane at a fixed distance, the radius, from a fixed point called the centre. A chord is a line segment joining two points of the circle. A chord through the centre is a diameter, the longest chord, with length twice the radius. An arc is a part of the circle. A secant is a line that cuts the circle at two points, and a tangent is a line that touches it at exactly one point, called the point of contact.
The distance of a chord from the centre means the length of the perpendicular from the centre to the chord. This perpendicular distance is the shortest distance from the centre to the chord.
Theorem 1 and its converse:
- A perpendicular from the centre of a circle to a chord bisects the chord.
- Converse: the line joining the centre to the mid point of a chord is perpendicular to the chord.
- Proof idea: join the centre O to the ends A and B of the chord. The two right triangles OMA and OMB have equal hypotenuses (radii) and a common side OM, so they are congruent, and AM = MB.
From the first theorem, the perpendicular bisector of any chord passes through the centre. So the centre can be found by drawing the perpendicular bisectors of two chords. Their meeting point is the centre. This is also how a circle is drawn through three points that are not on a line.
Theorem 2 and its converse:
- Equal chords of a circle are equidistant from the centre.
- Converse: chords that are equidistant from the centre are equal in length.
- A longer chord is nearer to the centre, and the diameter has distance 0.
These theorems give a useful right triangle. If a chord has length c, the radius is r and the distance of the chord from the centre is d, then r² = d² + (c/2)². Knowing two of the three quantities, the third can be found.
A tangent is perpendicular to the radius drawn to the point of contact. The converse is also true: a line drawn perpendicular to a radius at its outer end is a tangent. Through a point on a circle there is exactly one tangent. Through an external point there are exactly two tangents.
The two tangents drawn from an external point to a circle are equal in length. If P is an external point with tangents PA and PB, then PA = PB. The proof joins O to A, B and P and uses the right triangles OAP and OBP, which have a common hypotenuse OP and equal radii OA and OB. Also, OP bisects the angle APB and the angle AOB.
If P is at distance OP from the centre, the length of the tangent is PT = √(OP² − r²), by the Pythagorean theorem in the triangle OTP, where the angle at T is a right angle. In the quadrilateral OAPB, the angles at A and B are right angles, so ∠AOB + ∠APB = 180°.
The tangent chord angle theorem states that the angle between a tangent and a chord drawn through the point of contact is equal to the angle in the alternate segment. It is often used to find unknown angles in diagrams.
Two circles that touch each other:
- Touching externally: the distance between the centres is the sum of the radii, r₁ + r₂. The point of contact lies on the line of centres.
- Touching internally: the distance between the centres is the difference of the radii, r₁ − r₂.
- Circles that touch have a common tangent at the point of contact, perpendicular to the line of centres.
Study tip: Begin every theorem with a labelled figure, then write Given, To prove and Construction, followed by the proof in numbered steps with a reason for each. For numerical problems always look for a right triangle and use the Pythagorean theorem. Mark right angles and equal lengths on the diagram.
Definitions and flashcards
Say the meaning first, then open the card to check.
Chord
A line segment joining two points on a circle.
Diameter
A chord passing through the centre.
Secant
A line that meets a circle at two points.
Tangent
A line that touches a circle at exactly one point.
Point of contact
The single point where a tangent meets the circle.
Concentric circles
Circles that have the same centre.
Alternate segment
The segment of the circle on the opposite side of the chord from the angle.
Line of centres
The line passing through the centres of two circles.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Chord and distance | r² = d² + (c/2)² | r radius, d distance from centre, c chord length. |
| Tangent length | PT = √(OP² − r²) | P outside the circle at distance OP from the centre. |
| Tangent angle sum | ∠AOB + ∠APB = 180° | A and B points of contact of tangents from P. |
| Circles touching externally | O₁O₂ = r₁ + r₂ | Distance between centres. |
| Circles touching internally | O₁O₂ = r₁ − r₂ | r₁ is the larger radius. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
What is the difference between a secant and a tangent? (2 marks)
A secant cuts the circle at two points. A tangent touches the circle at only one point, the point of contact.
State the theorem about a perpendicular from the centre to a chord. (2 marks)
A perpendicular drawn from the centre of a circle to a chord bisects the chord.
State the converse of the theorem on the tangent and the radius. (2 marks)
A line drawn at right angles to a radius at its outer end, on the circle, is a tangent to the circle.
Prove that equal chords of a circle are equidistant from the centre. (5 marks)
Let AB and CD be equal chords of a circle with centre O. Draw OM perpendicular to AB and ON perpendicular to CD. By the first theorem M and N are mid points, so AM = AB/2 and CN = CD/2. Since AB = CD, AM = CN. In the right triangles OMA and ONC, the hypotenuses OA and OC are equal radii and AM = CN. Therefore the triangles are congruent (RHS) and OM = ON. So the chords are equidistant from the centre.
Prove that the tangents drawn from an external point to a circle are equal. (5 marks)
Let PA and PB be tangents from P to a circle with centre O. Join OA, OB and OP. A tangent is perpendicular to the radius at the point of contact, so ∠OAP = ∠OBP = 90°. In the right triangles OAP and OBP, OA = OB (radii) and OP is common, the hypotenuse. So the triangles are congruent (RHS) and PA = PB.
How many common tangents can two circles have in the cases of external and internal touching? (3 marks)
Two circles that touch externally have three common tangents: two outer ones and one inner tangent at the point of contact. Two circles that touch internally have only one common tangent, drawn at the point of contact.
Numericals with worked steps
A chord of length 16 cm is in a circle of radius 10 cm. Find its distance from the centre.
- Half the chord = 16/2 = 8 cm.
- d² = r² − 8² = 100 − 64 = 36.
- d = 6 cm.
Answer: 6 cm
A chord 12 cm long is 8 cm from the centre of a circle. Find the radius.
- Half the chord = 6 cm.
- r² = 8² + 6² = 64 + 36 = 100.
- r = 10 cm.
Answer: 10 cm
A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from P.
- The tangent is perpendicular to the radius at the point of contact T.
- PT² = OP² − OT² = 169 − 25 = 144.
- PT = 12 cm.
Answer: 12 cm
Tangents PA and PB are drawn from P to a circle with centre O and ∠APB = 50°. Find ∠AOB.
- ∠OAP = ∠OBP = 90° because a tangent is perpendicular to the radius.
- The angles of the quadrilateral OAPB sum to 360°.
- ∠AOB = 360° − 90° − 90° − 50° = 130°.
Answer: 130°
Two circles of radii 9 cm and 4 cm touch each other. Find the distance between their centres when they touch externally and when they touch internally.
- External: 9 + 4 = 13 cm.
- Internal: 9 − 4 = 5 cm.
Answer: 13 cm externally and 5 cm internally
Common mistakes
- Using the whole chord instead of half the chord in the right triangle with the radius.
- Forgetting that the radius to the point of contact is perpendicular to the tangent, so the right angle is missed.
- Adding the radii for circles that touch internally, or subtracting them for circles that touch externally.
- Writing the proof of congruence without naming the reason RHS or the common side.
- Drawing a diagram that is not clear, with the perpendicular not marked by a right angle sign.