Same arms, same chord
Equal angles at the centre give equal chords
Arms at the centre
Stand at the centre of a round floor and stretch your arms so they make a certain angle. Your fingertips touch the edge of the floor. The distance between your fingertips is a chord. Now the floor turns, and you keep your arms exactly as they are.
After the floor has turned, is the distance between your fingertips the same?
What this lesson covers
The idea
Chords of a circle that subtend equal angles at the centre are equal, because the triangles they form with the centre are congruent by SAS.
Arms at the centre
Stand at the centre of a round floor and stretch your arms so they make a certain angle. Your fingertips touch the edge of the floor. The distance between your fingertips is a chord. Now the floor turns, and you keep your arms exactly as they are.
After the floor has turned, is the distance between your fingertips the same?
- Yes, it is the same
- No, it changes as the floor turns
- It depends how far the floor turns
Copy the angle
Your arms make ∠AOB. Drag D and E round the circle until ∠DOE is the same angle. Then read the table.
Two sides and the angle
Chords of a circle that subtend equal angles at the centre are equal.
OA = OD and OB = OE, because all four are radii. The angle between each pair is the same: ∠AOB = ∠DOE. So triangles OAB and ODE are congruent by SAS, and AB = DE.
Notes
Chords of a circle that subtend equal angles at the centre are equal, because the triangles they form with the centre are congruent by SAS.
Check yourself
Chords AB and DE of a circle with centre O have ∠AOB = ∠DOE = 60°. AB = 5 cm. What is DE, in cm?
Answer: 5 cm
The chords subtend equal angles at the centre, so they are equal: DE = AB = 5 cm.
∠AOB = ∠DOE, where O is the centre of the circle. Why are triangles OAB and ODE congruent?
You stand at the centre of a round floor with your arms at 90° to each other. The floor turns, and you keep your arms as they are. What happens to the distance between your fingertips?
In a circle with centre O, ∠AOB = ∠COD. AB = (3x + 1) cm and CD = 13 cm. What is x?
Answer: 4
AB = CD, so 3x + 1 = 13. Then 3x = 12 and x = 4.
- All three sides are equal (SSS). We do not know that AB = DE yet. That is what we want to show.
- There is a right angle in each triangle (RHS). Nothing says the triangles have a right angle.
- Two radii and the angle between them are equal in each triangle (SAS) — correct. Yes. OA = OD and OB = OE are radii, and the angles between them are equal.
- It stays the same — correct. Yes. Your arms still make 90° at the centre, so the new chord is equal to the first.
- It becomes longer. The angle at the centre has not changed, so the chord has not changed either.
- It becomes shorter. The angle at the centre has not changed, so the chord has not changed either.