Equal angles at the centre give equal chords
A round cake is cut into slices with equal tip angles. Is the straight line across each slice's rim equally long?
- A circle with centre O.
- Chord AB makes 100° at O.
- Chord DE also makes 100° at O.
- OA, OB, OD and OE are radii. They are equal.
- Two sides and the angle between them match. So AB = DE.
The rule
Yes: in one circle, equal tip angles give equal chords. A slice's tip is the centre of the cake.
Join the ends of a chord to the centre O. The chord makes an angle at O.
Rule: In one circle, equal angles at the centre give equal chords.
Look at triangles AOB and DOE. OA = OD and OB = OE, because all four are radii. The angles at O match. Two sides and the angle between them match (SAS). So the triangles match, and AB = DE.
In one circle, equal chords make equal angles at the centre. The two radii and the chord make a triangle. Two equal chords give triangles that match by three equal sides. This works in the other direction too: equal angles make equal chords.
| If you know | Then |
|---|---|
| Equal chords | Equal angles at the centre |
| Equal angles at the centre | Equal chords |
| Equal chords | Equal distances from the centre |
| Equal distances from the centre | Equal chords |
A circle has radius 10 cm. Chord AB makes 90° at the centre. Triangle AOB has a right angle at O. AB is opposite that right angle. So AB² = 10² + 10² = 200, and AB is about 14.1 cm. Any other 90° chord in this circle is also about 14.1 cm.
Watch out!
The circles must be the same size. A 60° angle in a small circle gives a short chord. The same angle in a big circle gives a longer chord.
Read two facts about two chords. Are they equal?
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