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A line segment joining two points on a circumference of a circle is called a chord of a circle. A chord of a circle passing through the centre of a circle is called the diameter of a circle and it’s the longest chord of a circle.
Let’s understand what is a chord of a circle and its properties with examples.
What is a Chord of a Circle?
A line segment that joins two points on the circumference of the circle is called the chord of the circle. In other words, any line segment whose endpoints lie on the circumference of a circle is the chord of a circle.
In the above figure, $\text{AB}$ and $\text{DE}$ are chords as the endpoints of the line segments lie on the circumference of a circle. The chord $\text{DE}$ is also the diameter of the circle.
The line segment $\text{OC}$ is not a chord as only one endpoint lies on the circumference.
Note: A chord that passes through the centre of a circle is called the diameter of the circle.
Properties of the Chord of a Circle
The following are the important properties of a chord of a circle.
- The perpendicular to a chord, drawn from the centre of the circle, bisects the chord.
- The chords of a circle, equidistant from the centre of the circle are equal.
- When a chord of a circle is drawn, it divides the circle into two regions, referred to as the segments of the circle: the major segment and the minor segment.
- A chord when extended infinitely on both sides becomes a secant of a circle.
Chord Length Formula
There are two basic formulas to find the length of the chord of a circle. The first one is by using the Pythagoras theorem and the second one is by using trigonometry.
Using Pythagoras Theorem: Chord length using perpendicular distance from the centre = $2 \times \sqrt{r^2 – d^2}$.
Consider a circle with centre $\text{O}$ and the chord $\text{AB}$ and radius $\text{OA}$. The distance of the chord $\text{AB}$ from the centre is $\text{OC}$.
In right $\triangle \text{OAC}$, by Pythagoras theorem, $\text{AC} = \sqrt{\text{OA}^2 – \text{OC}^2}$
$=>\text{AC} = \sqrt{r^2 – d^2}$
Therefore, length of chord $\text{AB} = 2 \times \text{AC} = 2 \times \sqrt{r^2 – d^2}$
Using Trigonometry: Chord length using perpendicular distance from the centre = $2 \times r \times \sin \frac{\theta}{2}$.
Consider a circle with centre $\text{O}$ and the chord $\text{AB}$ and radius $\text{OA}$. The distance of the chord $\text{AB}$ from the centre is $\text{OC}$. Further, let the vertical $\angle \text{AOB} = \theta$.
Therefore, in right $\triangle \text{OAC}$, $\sin \frac{\theta}{2} = \frac{\text{AC}}{r}$
$=> \text{AB} = 2 \times r \times \sin \frac{\theta}{2}$
Theorems of Chord of a Circle
There are a few theorems based on the chord of a circle.
Theorem 1: Equal chords of a circle subtend equal angles at the centre.
Consider a circle with centre $\text{O}$ and two equal chords $\text{AB}$ and $\text{CD}$ and we want to prove that $\angle \text{AOB} = \angle \text{COD}$.
In triangles $\text{AOB}$ and $\text{COD}$,
$\text{OA} = \text{OC}$ (Radii of a circle)
$\text{OB} = \text{OD}$ (Radii of a circle)
$\text{AB} = \text{CD}$ (Given)
Therefore, $\triangle \text{AOB} \cong \triangle \text{COD}$ (SSS rule)
This gives $\angle \text{AOB} = \angle \text{COD}$ (Corresponding sides of congruent triangles)
Theorem 2: The perpendicular to a chord, drawn from the centre of the circle, bisects the chord.
Consider the above figure, where $\text{O}$ is the centre of a circle and $\text{AB}$ is the chord.
Further, let $\text{OP}$ be the perpendicular drawn on the chord $\text{AB}$ and we want to prove that $\text{AP} = \text{PB}$.
In $\triangle \text{OAP}$ and $\triangle \text{OBP}$
$\text{OA} = \text{OB}$ (radii of a circle)
$\text{OP} = \text{OP}$ (common side)
$\angle \text{OPA} = \angle \text{OPB} = 90^{\circ}$
By RHS rule, $\triangle \text{OAP} \cong \triangle \text{OBP}$
Therefore, $\text{AP} = \text{PB}$ (Corresponding sides of congruent triangles)
Thus, the perpendicular to a chord, drawn from the centre of the circle, bisects the chord.
Practice Problems
- Two equal chords AB and CD of a circle, when produced, intersect at a point P. Prove that PB = PD.
- Two circles of radii 5 cm and 3 cm intersect at two points, and the distance between their centres is 4 cm. Find the length of the common chord.
- The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance of 4 cm from the centre, what is the distance of the other chord from the centre?
FAQs
What is the chord of a circle?
The chord is a line segment that joins two points on the circumference of the circle. A chord only covers the part inside the circle.
What is the formula for chord length?
The length of any chord can be calculated using the formula Chord Length = $2 \times \sqrt{r^2 − d^2}$.
Is diameter a chord of a circle?
Yes, the diameter is also considered as a chord of the circle. The diameter is the longest chord possible in a circle and it divides the circle into two equal parts.
Conclusion
A line segment that joins two points on the circumference of the circle is called the chord of the circle. Among all the chords of a circle, the one that passes through the centre of a circle is the longest chord and is also called the diameter of a circle.
Recommended Reading
- What is a Circle – Parts, Properties & Examples
- How to Construct a Perpendicular Line (With Steps & Examples)
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
- Difference Between Axiom, Postulate and Theorem
- Lines in Geometry(Definition, Types & Examples)
- What Are 2D Shapes – Names, Definitions & Properties
- 3D Shapes – Definition, Properties & Types
