Circles: the perpendicular that halves a chord
Draw a chord across a round chautari platform at different distances from the centre. See that the perpendicular from the centre cuts it in half, and use Pythagoras to find how long it is.
Core concept
The perpendicular from the centre of a circle to a chord bisects the chord, so half the chord is √(r² − d²); the nearer the chord is to the centre, the longer it is, and through the centre it is a diameter
Topic: Circles
A circle has a centre O and a radius r, the distance from O to any point on the circle. A chord is a straight line joining two points on the circle; the longest chord passes through the centre and is a diameter, 2r long. Draw a perpendicular from O to a chord, d away from the centre. It always cuts the chord into two equal halves. The radius to one end of the chord, the perpendicular and half the chord make a right-angled triangle, so by Pythagoras, half the chord = √(r² − d²), and the whole chord is twice that. If d is as big as r or bigger, the line does not cut the circle at all.
What you will learn
- Name the centre, radius, diameter and chord of a circle.
- State that the perpendicular from the centre bisects a chord.
- Find the length of a chord from the radius and its distance from the centre.
- Explain why chords nearer the centre are longer.
Before you start
- Know Pythagoras' rule and how to find a square root.
Materials
- A circle drawn with a compass on paper, or a round plate
Equipment
- A compass, a ruler and a set square
Procedure
- Record a chord 5 cm from the centre of a circle of radius 13 cm.
- Move it to 12 cm from the centre and record.
- Put it through the centre and record.
- Try a chord 6 cm from the centre of a 5 cm circle.
How this simulation is simplified
- The chord is a straight line, measured in whole centimetres from the centre.