Chords of a Circle Theorems



In this lesson, 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.

perpendicular bisector chord

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.

chords center 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.

Geometry Help: Circles, Radius Chord relationships, distance from the center to a chord
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.

congruent chords 

If PQ = RS then OA = OB or

If OA = OB then PQ = RS

 

This video defines and shows how to use the Chords Equidistant from the Center of a Circle Theorem. 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.

This video explains the theorem on chords and arcs and shows an example on how to use theorem. 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.

This video discusses the following theorems:
(1) Congruent central angles have congruent chords
(2) Congruent chords have congruent arcs
(3) Congruent arcs have congruent central angles

This video describes the four properties of chords.
(1) If two chords in a circle are congruent, then they determine two cental 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 cm

Using Pythagoras’ theorem,

Since OQ is a radius that is perpendicular to the chord RS, it divides the chord into two equal parts.

RS = 2RP = 2 × 3 = 6 cm

 

The following video gives another example of how the theorem - If a diameter is perpendicular to a chord, then it bisects the chord and its arc - can be used.







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