Middle school maths, year 3 · unit 6
Properties of Circles
To measure the diameter of a circle drawn on the playground, the string has to pass through the centre. But how do you know it does? A circle has several properties that always hold. The perpendicular from the centre cuts a chord in half, and from any point on the circle the same arc is seen at the same angle. We start by laying one stick across a hoop.
Chords and the centre
A board to adjustWhat happens to the length of a chord as it moves away from the centre? Drag the stick up and down.
Tangents
A board to adjustSome lines meet a circle at exactly one point. Slide the stick to make one, then compare the two tangents from a point outside the circle.
Inscribed angles
A board to adjustDoes the angle at a point on the circle, looking at an arc, change as the point moves? Drag point P round the circle.
Cyclic quadrilaterals
A board to adjustWhen all four vertices lie on a circle, opposite angles are tied together. Drag point D.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- The perpendicular from the centre of a circle to a chord bisects the chord. The perpendicular bisector of a chord passes through the centre.
- In one circle, two chords the same distance from the centre are equally long. The nearer the centre, the longer the chord, and the longest chord is the diameter.
- A tangent is perpendicular to the radius through the point of contact. The two tangents from a point outside a circle are equally long.
- Inscribed angle = central angle ÷ 2. So all inscribed angles on the same arc are equal.
- The angle in a semicircle is 90°. A triangle with a diameter as one side and its third vertex on the circle is right-angled.
- In a cyclic quadrilateral, opposite angles add up to 180°.