Trigonometric Ratios – SS2 Mathematics Lesson Note (Second Term)

Trigonometric ratios can be used to find the length of a chord of a circle. When the angles are not given, Pythagoras’ theorem is used instead. It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, that is c² = a² + b².

Behavioural objectives

By the end of the lesson, students should be able to:

• state Pythagoras’ theorem;
• explain when Pythagoras’ theorem is used instead of trigonometric ratios to find the length of a chord;
• sketch and label a right-angled triangle from a chord and the centre of a circle;
• calculate the length of a chord, or the distance from the centre to a chord, using Pythagoras’ theorem.

How do we find the length of a chord when angles are not given?

Trigonometric ratios can be used to find the length of chords of a given circle. In some cases, however, the angles are not given. Pythagoras’ theorem is used to find the lengths of chords in such cases.

What is Pythagoras’ theorem?

Pythagoras’ theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

c² = a² + b²

Examples

1. A chord is drawn 3 cm away from the centre of a circle of radius 5 cm. Calculate the length of the chord.
2. In a figure, O is the centre of the circle HKL. HK = 16 cm, HL = 10 cm and the perpendicular from O to HK is 4 cm. What is the length of the perpendicular from O to HL?

Solution to example 1

Sketch a right-angled triangle and label it correctly. Let AC be the chord, O the centre and B the midpoint of AC, so that OB = 3 cm and OA = 5 cm.

|AB|² + 3² = 5² (Pythagoras’ theorem)

|AB|² = 5² – 3²

|AB|² = 25 – 9

|AB|² = 16

AB = √16 = 4 cm

Since B is the midpoint of chord AC, then:

Length of chord AC = 2 × AB = 2 × 4 cm = 8 cm

Solution to example 2

Let the distance from O to HL = x cm.

In right-angled triangle OMH (M is the midpoint of HK, so HM = 8 cm and MO = 4 cm):

|OH|² = |HM|² + |MO|²

|OH|² = 8² + 4²

= 64 + 16

= 80

∴ |OH| = √80 cm

But OH is a radius of the circle, that is, r = OH = OL = √80 cm.

In right-angled triangle ONL (N is the midpoint of HL, so NL = 5 cm):

OL² = ON² + NL²

(√80)² = x² + 5²

80 – 25 = x²

55 = x²

Take the square root of both sides:

√55 = x = 7.416 cm (to 3 decimal places)

∴ The length of the perpendicular from O to HL is 7.416 cm.

Evaluation

1. State Pythagoras’ theorem.
2. When is Pythagoras’ theorem used to find the length of a chord instead of trigonometric ratios?
3. In example 1, why is the length of the chord equal to 2 × AB?
4. A chord is drawn 6 cm away from the centre of a circle of radius 10 cm. Calculate the length of the chord.

Assignment

A chord is drawn 5 cm away from the centre of a circle of radius 13 cm. Calculate the length of the chord.