Year 11 Standard · Unit 2 Right-angled triangles and circle geometry · Lesson 6 of 8

Why the radius meets a chord and a tangent at a right angle

Every radius of a circle has the same length, and that single fact proves three results about chords and tangents. This lesson proves them, then uses them with Pythagoras.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: a right-angled triangle has hypotenuse 41 and one leg 9. Find the other leg.
    Answer40, because 41² − 9² = 1681 − 81 = 1600.
  2. From Year 7: what is the angle sum of a quadrilateral?
    Answer360°.
  3. From Year 9: two sides of a triangle are equal. What can you say about the angles opposite them?
    AnswerThey are equal: an isosceles triangle.
Words used in this lesson
chord
a straight line joining two points on a circle. For example: the bar across a window.
tangent
a line that touches a circle at exactly one point. For example: a ruler laid against a wheel.
radius
a line from the centre to the circle, or its length. For example: a spoke of a wheel.
bisect
cut exactly in half. For example: the line from the centre to the middle of a chord.
perpendicular
at a right angle to. For example: the radius at the tangent.
point of contact
the one point where a tangent touches the circle. For example: T in the figure.

A bar across a window

A circular window has a radius of 17 cm. A straight bar is fixed across it, with its two ends on the frame, and the bar is 8 cm from the centre of the window.

Predict first: is the bar longer or shorter than the window's radius, 17 cm? Write down a guess and a reason, then use the first figure and worked example 2 to see what decides it.

8
3
√55 ≈ 7.42half the chord
14.83chord length

Radius 8, distance 3: half the chord is √(8² − 3²) = √55 ≈ 7.42, so the chord is ≈ 14.83.

Show me the steps
  1. Radius 8, distance 3: half the chord is √(64 − 9) = √55 ≈ 7.42, so the chord is ≈ 14.83.
  2. Make the distance 0: the chord goes through the centre. Half is √64 = 8 and the chord is 16, a diameter, the longest chord.
  3. Make the distance 7: half the chord is √(64 − 49) = √15 ≈ 3.87, so the chord is ≈ 7.75. Further from the centre means shorter.
  4. Make the distance 8: half the chord is 0, so the line touches the circle at one point. It is a tangent, and the radius to that point is perpendicular to it.

Three results, each from the fact that all radii are equal.

1. The perpendicular from the centre bisects a chord. Let M be the foot of the perpendicular from O to the chord AB. Triangles OMA and OMB have right angles at M, and OA = OB because both are radii. By Pythagoras AM² = OA² − OM² = OB² − OM² = MB², so AM = MB. (Worked example 1 proves the converse: the line from O to the midpoint is perpendicular to the chord.)

2. A tangent is perpendicular to the radius at the point of contact. Suppose it were not. Drop the perpendicular from O to the tangent line, with foot F. F is not T. Mark the point Q on the line on the other side of F, with FQ = FT. Triangles OFT and OFQ have a right angle at F, the common side OF and FT = FQ, so they are congruent and OQ = OT = r. Then Q is a second point where the line meets the circle, but a tangent meets it only at T: a contradiction.

3. Two tangents from a point are equal. In the right-angled triangles OTP and OT'P, OT = OT' = r and OP is common, so PT² = OP² − r² = PT'², and PT = PT'.

4
11
√105 ≈ 10.25each tangent length
42.6°angle at P between the tangents

Radius 4, OP = 11: each tangent is √(11² − 4²) = √105 ≈ 10.25. The angle at P is 42.6° and the angle TOT' is 137.4°; they sum to 180°.

Show me the steps
  1. Radius 4, OP = 11: each tangent is √(121 − 16) = √105 ≈ 10.25. The angle at P is 42.6° and the angle at the centre is 137.4°, which sum to 180°.
  2. Make OP = 16 with radius 4: each tangent is √(256 − 16) = √240 ≈ 15.49 and the angle at P shrinks to 29.0°. A point far away sees the circle in a narrow cone.
  3. Make the radius 8 and OP = 9: each tangent is √(81 − 64) = √17 ≈ 4.12 and the angle at P is 125.5°. A point close to the circle sees it wide.
  4. In every setting the two tangents are equal in length, because they are the legs of two right-angled triangles with the same hypotenuse and the same radius.

Find the hypotenuse before you add or subtract. In triangle OTP the longest side is OP, so PT² = OP² minus r², never plus. With OP = 41 and r = 9, PT is 40, not √(41² + 9²). Also, the chord is twice the leg of the triangle, so halve it before using Pythagoras.

Worked examples

1. Prove that the line from the centre of a circle to the midpoint of a chord is perpendicular to the chord.

  1. Let O be the centre and M the midpoint of chord AB. OA = OB because both are radii.
  2. AM = MB because M is the midpoint, and OM is common to both triangles.
  3. So triangles OMA and OMB are congruent (three equal sides), and angle OMA = angle OMB.
  4. The two angles lie on a straight line, so they sum to 180°. Each is 90°.

2. The window bar. The radius is 17 cm and the bar is 8 cm from the centre. How long is the bar?

  1. Half the bar is a leg of a right-angled triangle with hypotenuse 17 and other leg 8.
  2. Half the bar = √(17² − 8²) = √225 = 15. The bar is 30 cm, longer than the 17 cm radius.

3. A circle has radius 29 cm and a chord of length 40 cm. How far is the chord from the centre?

  1. Half the chord is 20, so d² = 29² − 20² = 841 − 400 = 441.
  2. d = 21 cm.

Your go. The first step is given. A tangent from a point P touches a circle of radius 33 cm, centre O, at T. OP = 65 cm. How long is PT?

  1. The tangent is perpendicular to the radius, so triangle OTP has a right angle at T, with hypotenuse OP.
  2. PT² = OP² − 33². Work out PT.
Check the answer

PT² = 65² − 33² = 4225 − 1089 = 3136, so PT = 56 cm. Adding the squares instead would give √5314, which is more than OP, and a leg can never be longer than the hypotenuse.

Check before you go on

1. O is the centre of a circle, AB is a chord and M is its midpoint. Which statement gives the angle OMA and a correct reason?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A chord is 12 cm from the centre of a circle of radius 37 cm. How long is the chord, in cm?

3. A tangent from a point P touches a circle of radius 9 cm at T. The centre O is 15 cm from P. How long is the tangent PT, in cm?

Level 2 · Building

4. A chord of length 24 cm is drawn in a circle of radius 13 cm. How far is the chord from the centre, in cm?

5. Two tangents from P touch a circle of radius 6 cm, centre O, at A and B. OP = 10 cm. What is the area of the quadrilateral OAPB, in cm²?

6. Two tangents from P touch a circle, centre O, at A and B. The angle APB is 50°. What is the angle AOB, in degrees?

Level 3 · Stretch

7. Two circles share a centre. Their radii are 13 cm and 5 cm. A chord of the larger circle touches the smaller circle. How long is the chord, in cm?

8. A circle has radius 5 cm. A chord is d cm from the centre, and its length is (2d + 2) cm. Find d.

Something to think about

A chord is 12 cm long in a circle of radius 10 cm. Without a calculator, how far is it from the centre? Then say what happens to the distance if the chord grows towards 20 cm.

One way to think about it

The half chord is 6, so d² = 100 − 36 = 64 and d = 8 cm. As the chord grows towards 20 cm, the half chord approaches 10 and d² = 100 − 100 = 0, so the chord moves to the centre and becomes a diameter.

Exit ticket

9. Two tangents from a point P to a circle have lengths PA = (2x + 3) cm and PB = (5x − 6) cm. What is x?