Same thread, same angle
Equal chords subtend equal angles at the centre
A thread on a wheel
Tie a thread tight between two points of a wheel. The thread is a chord. Join its ends to the centre: the thread subtends an angle at the centre. Now spin the wheel, so the thread moves to a new position.
Two chords of a circle are equal in length. What about the angles they subtend at the centre?
What this lesson covers
The idea
Equal chords of a circle subtend equal angles at the centre, because the triangles they form with the centre are congruent by SSS.
A thread on a wheel
Tie a thread tight between two points of a wheel. The thread is a chord. Join its ends to the centre: the thread subtends an angle at the centre. Now spin the wheel, so the thread moves to a new position.
Two chords of a circle are equal in length. What about the angles they subtend at the centre?
- The angles are equal too
- The angles are different
- Only the chord nearer the top has the bigger angle
Match the chords
Chord AB is fixed. Drag D and E round the circle until the chord DE is as long as AB. Then read the table.
Congruent triangles
Equal chords of a circle subtend equal angles at the centre.
Join the ends of both chords to the centre O. The four radii are equal and AB = DE, so triangles OAB and ODE have three pairs of equal sides. They are congruent by SSS, so ∠AOB = ∠DOE.
Notes
Equal chords of a circle subtend equal angles at the centre, because the triangles they form with the centre are congruent by SSS.
Check yourself
Chords AB and DE of a circle with centre O are equal. AB subtends 70° at O. What angle does DE subtend at O, in degrees?
Answer: 70 °
Equal chords subtend equal angles at the centre, so ∠DOE = ∠AOB = 70°.
Chords AB and DE of a circle with centre O are equal. Why are triangles OAB and ODE congruent?
A thread 6 cm long is tied between two points of a wheel. The wheel is spun, so the thread lies along a new chord. What happens to the angle the thread subtends at the centre?
Equal chords AB and DE of a circle with centre O. ∠AOB = 2x° and ∠DOE = 90°. What is x?
Answer: 45
Equal chords subtend equal angles, so 2x = 90 and x = 45.
- Two sides and the angle between them are equal (SAS). We do not know the angles yet. That is what we want to show.
- Two angles and a side are equal (ASA). We know no angles yet, only the sides.
- All three sides are equal: OA = OD, OB = OE and AB = DE (SSS) — correct. Yes. OA, OB, OD and OE are all radii, and AB = DE is given.
- It stays the same — correct. Yes. The new chord is also 6 cm long, and equal chords subtend equal angles at the centre.
- It becomes larger. The new chord is the same length, so it subtends the same angle.
- It becomes smaller. The new chord is the same length, so it subtends the same angle.