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Cambridge IGCSE0580

Circle theorems II

Mathematics 0580 Chapter Notes

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Circle theorems II
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Circle theorems II notes

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In this chapter

  1. 1The perpendicular bisector of a chord
  2. 2Equal chords
  3. 3Tangents from an external point
  4. 4Giving geometrical reasons

1. The perpendicular bisector of a chord

The perpendicular bisector of a chord passes through the centre of the circle.

So a line from the centre to the midpoint of a chord meets the chord at 90°. Two radii and the chord make an isosceles triangle. The line from the centre splits it into two right-angled triangles.

Take a circle of radius r and a chord of length c at a distance d from the centre. The radius is the hypotenuse. Use Pythagoras with half the chord.

r² = d² + (c/2)²

Examiner insight

“This question also proved to be very demanding as most candidates did not appreciate that the perpendicular from O to the chord bisected the chord and consequently they did not draw an appropriate right-angled triangle in which to use Pythagoras’ theorem.” “Those that identified the perpendicular from O to AB as 18 – r usually went on to make a correct statement for Pythagoras’ theorem.”

Worked example 14 marks

Do not use a calculator. A circle has centre O and radius 13 cm. The chord AB has length 24 cm. Find the distance from O to AB.

  1. 1

    the perpendicular from O to AB bisects AB at M, so OM is the distance and ∠ OMA = 90^(°) [1]

  2. 2

    AM = 24 ÷ 2 = 12 cm [1]

  3. 3

    OM² = 13² - 12² = 169 - 144 = 25 [1]

  4. 4

    OM = 5 cm [1]

2. Equal chords

Equal chords are equidistant from the centre. The distance from the centre to a chord is the perpendicular distance.

If chords AB and CD are equal, the perpendicular from O to AB is the same length as the perpendicular from O to CD.

Find the distance to one chord. The other chord is the same distance away.

Examiner insight

“Common incorrect values used for the radius came from the length, or half the length, of the chord PQ.”

Worked example 14 marks

Do not use a calculator. In a circle with centre O, the chords PQ and RS are equal. PQ = 30 cm. The distance from O to PQ is 8 cm.

(a) Write down the distance from O to RS.

(b) Find the radius of the circle.

  1. 1

    (a) equal chords are equidistant from the centre: 8 cm [1]

  2. 2

    (b) half the chord = 30 ÷ 2 = 15 cm [1]

  3. 3

    r² = 8² + 15² = 64 + 225 = 289 [1]

  4. 4

    r = 17 cm [1]

3. Tangents from an external point

A point outside the circle is an external point.

Two tangents from an external point are equal in length. If tangents from T touch the circle at P and Q, then TP = TQ. The reason must say that the tangents come from the same point.

The angle between a tangent and the radius at the point of contact is 90°. This gives a right-angled triangle to work in. It is not the reason why the tangents are equal.

Examiner insight

“Some good explanations from candidates included the words ‘tangents’ and ‘equal’ but did not mention the ‘common point’.” “Some gave the angle between radius and tangent = 90° as the reason here.”

Worked example 14 marks

Do not use a calculator. T is a point outside a circle with centre O and radius 6 cm. The tangents from T touch the circle at P and Q. OT = 10 cm. Find the length of TQ.

  1. 1

    the angle between tangent TP and radius OP is 90^(°) [1]

  2. 2

    TP² = 10² - 6² = 100 - 36 = 64 [1]

  3. 3

    TP = 8 cm [1]

  4. 4

    TQ = TP = 8 cm, because tangents from an external point are equal in length [1]

4. Giving geometrical reasons

A reason is a geometrical property, named in words. A calculation is not a reason.

Name the lines correctly: radius, chord, tangent. Write one short reason beside each step.

The three properties of this topic: the perpendicular bisector of a chord passes through the centre, equal chords are equidistant from the centre, and tangents from an external point are equal in length.

Examiner insight

“For some parts, candidates opted to show their calculations which are not acceptable as reasons.” “Some attempted to use the isosceles triangle and tangent properties but gave inadequate reasons such as the tangent is perpendicular to the centre.”

Worked example 14 marks

Do not use a calculator. The tangents from a point T touch a circle at A and B. Angle ATB = 50°. Find angle TAB. Give a reason for each step.

  1. 1

    TA = TB, because tangents from an external point are equal in length [1]

  2. 2

    triangle TAB is isosceles, so the base angles TAB and TBA are equal [1]

  3. 3

    ∠ TAB = (180^(°) - 50^(°)) ÷ 2, because the angles in a triangle add to 180^(°) [1]

  4. 4

    ∠ TAB = 65^(°) [1]

Recap

  • Extended: the perpendicular bisector of a chord passes through the centre.
  • Extended: the line from the centre to the midpoint of a chord meets it at 90°.
  • Extended: equal chords are equidistant from the centre.
  • Extended: tangents from an external point are equal in length.
  • Use half the chord in Pythagoras: r² = d² + (c/2)².
  • Give a named property as the reason. A calculation is not a reason.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. The perpendicular bisector of a chord of a circle is drawn. Write down the point that this line must pass through.1 mark
  2. The tangents TP and TQ are drawn to a circle from an external point T. Write down the relationship between TP and TQ and give the reason.2 marks
  3. Do not use a calculator. The chords AB and CD of a circle with centre O are equal in length. The distance from O to AB is 6 cm. Write down the distance from O to CD and give a reason.2 marks
  4. Do not use a calculator. A circle has centre O and radius 5 cm. The chord AB has length 8 cm. Find the distance from O to AB.3 marks
  5. Do not use a calculator. In a circle with centre O, M is the midpoint of the chord AB. Angle AOM = 50°. Find angle OAM. Give a reason for each step.3 marks
  6. A circle has centre O and radius 7.5 cm. The chord PQ has length 10 cm. Calculate the distance from O to PQ. Give your answer correct to 3 significant figures.3 marks
  7. Do not use a calculator. The tangents from a point T touch a circle at A and B. The circle has centre O and radius 12 cm. OT = 13 cm. Find the length of TB.3 marks
  8. The tangents from a point T touch a circle at A and B. The circle has centre O and radius 6.5 cm. OT = 15 cm. Calculate the perimeter of the quadrilateral OATB. Give your answer correct to 3 significant figures.4 marks
  9. Do not use a calculator. The cross-section of a tunnel is part of a circle with centre O and radius r m. The flat floor AB is a chord of length 6 m. The highest point of the tunnel is 5 m above the floor. O is above the floor. Find the value of r.5 marks
  10. A circular metal plate has centre O and radius 9 cm. Two straight parallel cuts are made across the plate. Each cut is a chord. The cuts have lengths 14 cm and 10 cm. Calculate the two possible distances between the cuts. Give each answer correct to 3 significant figures.6 marks

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