In these lessons, we will learn theorems that involve chords of a circle.

- Perpendicular bisector of a chord passes through the center of a circle.
- Congruent chords are equidistant from the center of a circle.
- If two chords in a circle are congruent, then their intercepted arcs are congruent.
- If two chords in a circle are congruent, then they determine two central angles that are congruent.

A **chord** is a straight line joining 2 points on the circumference of a circle.

**Theorem: A radius or diameter that is perpendicular to a chord divides the chord into two equal parts and vice versa. **

In the above circle, if the radius *OB* is perpendicular to the chord *PQ* then *PA* = *AQ*.

Converse: The perpendicular bisector of a chord passes through the center of a circle.

In the above circle, *OA* is the perpendicular bisector of the chord *PQ *and it passes through the center of the circle. *OB* is the perpendicular bisector of the chord *RS *and it passes through the center of the circle.

We can use this property to find the center of any given circle.

*Example**:*

Determine the centre of the following circle.

*Solution:*

**Step 1**: Draw 2 non-parallel chords

**Step 2: **Construct perpendicular bisectors for both the chords. The centre of the circle is the point of intersection of the perpendicular bisectors.

This video shows how to define a chord; how to describe the effect of a perpendicular bisector of a chord and the distance from the center of the circle.

The perpendicular bisector of a chord passes through the center of the circle.

Theorem: Congruent Chords are equidistant from the center of a circle.

Converse: Chords equidistant from the center of a circle are congruent.

If

PQ = RSthenOA = OBorIf

OA = OBthenPQ = RS

The theorem states:

(1) Chords equidistant from the center of a circle are congruent

(2) Congruent chords are equidistant from the center of a circle.

Theorem: If two chords in a circle are congruent then their intercepted arcs are congruent.

Converse: If two arcs are congruent then their corresponding chords are congruent.

It also shows the perpendicular bisector theorem.

(1) If a diameter or radius is perpendicular to a chord, then it bisects the chord and its arc.

(2) If two chords are congruent, then their corresponding arcs are congruent.

(3) If a diameter or radius is perpendicular to a chord, then it bisects the chord and its arc.

(4) In the same circle or congruent circle, two chords are congruent if and only if they are equidistant from the center. Theorem: If two chords in a circle are congruent then they determine two central angles that are congruent.

(1) Congruent central angles have congruent chords

(2) Congruent chords have congruent arcs

(3) Congruent arcs have congruent central angles

(1) If two chords in a circle are congruent, then they determine two central angles that are congruent.

(2) If two chords in a circle are congruent, then their intercepted arcs are congruent.

(3) If two chords in a circle are congruent, then they are equidistant from the center of the circle.

(4) The perpendicular from the center of the circle to a chord bisects the chord.

*Example: *

The figure is a circle with centre *O*. Given *PQ* = 12 cm. Find the length of *PA. *

* Solution: *

The radius *OB* is perpendicular to *PQ*. So, *OB* is a perpendicular bisector of *PQ*.

* Example: *

The figure is a circle with centre *O* and diameter 10 cm. *PQ* = 1 cm. Find the length of *RS*.

* Solution: *

OP=OQ – PQ

=5 cm – 1 cm = 4 cmUsing Pythagoras’ theorem,

Since

OQis a radius that is perpendicular to the chordRS, it divides the chord into two equal parts.

RS= 2RP= 2 × 3 = 6 cm

Example:

Find the length of the radius of a circle if a chord of the circle has a length of 12 cm and is 4 cm from the center of the circle.

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