I’m Up and Down, and Round and Round · Lesson 3 of 8
Chords and the Angles They Subtend
“Chords pull angles into the conversation, while the centre keeps score.”
• Understand the angle subtended by a chord at the centre. • Prove that equal chords subtend equal central angles. • Prove the converse using congruent triangles. • Recognise the isosceles triangle formed by two radii and a chord. • Use SSS and SAS congruence in circle proofs.
A chord is a line segment that joins any two points on a circle, but it also helps us understand an important relationship between the circle and its centre. If we join both endpoints of the chord to the centre of the circle, the two new line segments are radii. Since all radii of the same circle are equal, these two radii(plural of radius) form two equal sides of a triangle. Therefore, the triangle formed by the chord and the two radii is an isosceles triangle.
The angle formed at the centre between these two radii is called the angle subtended by the chord at the centre. In other words, a chord determines an angle at the centre of the circle. The size of this angle depends on the position and length of the chord, and this idea becomes very useful when we study the relationships between chords, arcs, and angles in a circle.
Because OA and OB are radii of the same circle, OA=OB. Therefore triangle OAB is isosceles.
Equal Chords Give Equal Central Angles
Equal chords of a circle subtend equal angles at the centre.
Suppose AB and DE are equal chords in the same circle with centre O. Join A, B, D and E to O. We compare triangles OAB and ODE.
| In ΔOAB and ΔODE | Reason |
|---|---|
| OA=OD | Radii |
| OB=OE | Radii |
| AB=DE | Given |
All three corresponding sides are equal, so the triangles are congruent by SSS. Corresponding central angles are therefore equal.
The Converse
Chords of a circle that subtend equal angles at the centre are equal.
Now suppose ∠AOB=∠DOE. Again compare triangles OAB and ODE. Two pairs of corresponding sides are radii and therefore equal, and the included central angles are equal. So the triangles are congruent by SAS.
Why Congruence Is the Natural Tool
A circle gives us many equal radii automatically. This often creates triangles with several equal sides. When we want to prove equality of angles or chords, triangle congruence converts those radius equalities into the desired result.
Problem
In a circle with centre O, chords AB and CD are equal. If ∠AOB=72°, find ∠COD.
- 1.Equal chords subtend equal angles at the centre.
- 2.AB=CD.
- 3.Therefore ∠AOB=∠COD.
- 4.So ∠COD=72°.
Practice Problems
- Show that the triangle formed by a chord and the centre of the circle is isosceles.
- Two equal chords AB and CD are drawn in the same circle. Prove that triangles OAB and OCD are congruent.
- If two central angles are each 84°, what can you say about their corresponding chords? Explain.
- A chord AB subtends 65° at the centre. Another chord CD is equal to AB. Find the angle subtended by CD at the centre.
- Explain why SSS is used in the proof 'equal chords imply equal angles' while SAS is used for the converse.
Key Takeaways
• A chord and two radii form an isosceles triangle. • Equal chords subtend equal angles at the centre. • Equal central angles subtend equal chords. • SSS proves the first theorem; SAS proves the converse. • Equal radii are the key structural fact behind both proofs.
Next, we connect the centre of a circle with the midpoint and perpendicular bisector of a chord.