What is the chord that passes through the center of a circle?

Related Pages
Circles
Tangent Of A Circle
Chords Of A Circle

Índice Show

Parts Of A Circle, Including Radius, Chord, Diameter, Central Angle, Arc, And Sector
Chord Length Formula
Example Question Using Chord Length Formula
Video Related to Chords
Chord of a Circle Theorems
Theorem 1: Equal Chords Equal Angles Theorem
Theorem 2: Equal Angles Equal Chords Theorem (Converse of Theorem 1)
Theorem 3: Equal Chords Equidistant from Center Theorem
Solved Examples
Practice Problems
More Topics Related to Chord and Chord Length of Circles

The following figures show the different parts of a circle: tangent, chord, radius, diameter, minor arc, major arc, minor segment, major segment, minor sector, major sector. Scroll down the page for more examples and explanations.

In geometry, a circle is a closed curve formed by a set of points on a plane that are the same distance from its center O. That distance is known as the radius of the circle.

Diameter

The diameter of a circle is a line segment that passes through the center of the circle and has its endpoints on the circle. All the diameters of the same circle have the same length.

Chord

A chord is a line segment with both endpoints on the circle. The diameter is a special chord that passes through the center of the circle. The diameter would be the longest chord in the circle.

Radius

The radius of the circle is a line segment from the center of the circle to a point on the circle. The plural of radius is radii.

In the above diagram, O is the center of the circle and

and

are radii of the circle. The radii of a circle are all the same length. The radius is half the length of the diameter.

Arc

An arc is a part of a circle.

In the diagram above, the part of the circle from B to C forms an arc.

An arc can be measured in degrees.

In the circle above, arc BC is equal to the ∠BOC that is 45°.

Tangent

A tangent is a line that touches a circle at only one point. A tangent is perpendicular to the radius at the point of contact. The point of tangency is where a tangent line touches the circle.

In the above diagram, the line containing the points B and C is a tangent to the circle.

It touches the circle at point B and is perpendicular to the radius

. Point B is called the point of tangency.

is perpendicular to

i.e.

The following video gives the definitions of a circle, a radius, a chord, a diameter, secant, secant line, tangent, congruent circles, concentric circles, and intersecting circles.

A secant line intersects the circle in two points.

A tangent is a line that intersects the circle at one point.

A point of tangency is where a tangent line touches or intersects the circle.

Congruent circles are circles that have the same radius but different centers.

Concentric circles are two circles that have the same center, but a different radii.

Intersecting Circles: Two circles may intersect at two points or at one point. If they intersect at one point then they can either be externally tangent or internally tangent.

Two circles that do not intersect can either have a common external tangent or common internal tangent.
In the common external tangent, the tangent does not cross between the two circles.
In the common internal tangent, the tangent crosses between the two circles.

Show Video Lesson

Parts Of A Circle, Including Radius, Chord, Diameter, Central Angle, Arc, And Sector

Show Video Lesson

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The chord of a circle can be defined as the line segment joining any two points on the circumference of the circle. It should be noted that the diameter is the longest chord of a circle which passes through the center of the circle. The figure below depicts a circle and its chord.

In the given circle with ‘O’ as the center, AB represents the diameter of the circle (longest chord), ‘OE’ denotes the radius of the circle and CD represents a chord of the circle.

Let us consider the chord CD of the circle and two points P and Q anywhere on the circumference of the circle except the chord as shown in the figure below. If the endpoints of the chord CD are joined to the point P, then the angle ∠CPD is known as the angle subtended by the chord CD at point P. The angle ∠CQD is the angle subtended by chord CD at Q. The angle ∠COD is the angle subtended by chord CD at the center O.

Chord Length Formula

There are two basic formulas to find the length of the chord of a circle which are:

Formula to Calculate Length of a Chord
Chord Length Using Perpendicular Distance from the Center	Chord Length = 2 × √(r2 − d2)
Chord Length Using Trigonometry	Chord Length = 2 × r × sin(c/2)

Where,

r is the radius of the circle
c is the angle subtended at the center by the chord
d is the perpendicular distance from the chord to the circle center

Example Question Using Chord Length Formula

Question: Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance from the chord to the center is 4 cm.

Solution:

Given radius, r = 7 cm

and distance, d = 4 cm

Chord length = 2√(r2−d2)

⇒ Chord length = 2√(72−42)

⇒ Chord length = 2√(49−16)

⇒ Chord length = 2√33

⇒ Chord length = 2×5.744

Or , chord length = 11.48 cm

Chord of a Circle Theorems

If we try to establish a relationship between different chords and the angle subtended by them in the center of the circle, we see that the longer chord subtends a greater angle at the center. Similarly, two chords of equal length subtend equal angle at the center. Let us try to prove this statement.

Theorem 1: Equal Chords Equal Angles Theorem

Statement: Chords which are equal in length subtend equal angles at the center of the circle.

Proof:

From fig. 3, In ∆AOB and ∆POQ

S.No.	Statement	Reason
1.	AB=PQ	Chords of equal length (Given)
2.	OA = OB = OP = OQ	Radius of the same circle
3.	△AOB ≅ △POQ	SSS axiom of Congruence
4.	∠AOB = ∠POQ	By CPCT from statement 3

Note: CPCT stands for congruent parts of congruent triangles.

The converse of theorem 1 also holds true, which states that if two angles subtended by two chords at the center are equal then the chords are of equal length. From fig. 3, if ∠AOB =∠POQ, then AB=PQ. Let us try to prove this statement.

Theorem 2: Equal Angles Equal Chords Theorem (Converse of Theorem 1)

Statement: If the angles subtended by the chords of a circle are equal in measure, then the length of the chords is equal.

Proof:

From fig. 4, In ∆AOB and ∆POQ

S.No.	Statement	Reason
1.	∠AOB = ∠POQ	Equal angle subtended at centre O (Given)
2.	OA = OB = OP = OQ	Radii of the same circle
3.	△AOB ≅ △POQ	SAS axiom of Congruence
4.	AB = PQ	From Statement 3 (CPCT)

Theorem 3: Equal Chords Equidistant from Center Theorem

Statement: Equal chords of a circle are equidistant from the center of the circle.

Proof:

Given: Chords AB and CD are equal in length.

Construction: Join A and C with centre O and drop perpendiculars from O to the chords AB and CD.

S.No.	Statement	Reason
1	AP = AB/2, CQ = CD/2	The perpendicular from centre bisects the chord
In △OAP and △OCQ
2	∠1 = ∠2 = 90°	OP⊥AB and OQ⊥CD
3	OA = OC	Radii of the same circle
4	OP = OQ	Given
5	△OPB ≅ △OQD	R.H.S. Axiom of Congruency
6	AP = CQ	Corresponding parts of congruent triangle
7	AB = CD	From statement (1) and (6)

Solved Examples

Example 1:

A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in the major segment.

Solution:

Let O be the centre, and AB be the chord of the circle.

So, OA and OB be the radii.

Given that chord of a circle is equal to the radius.

AB = OA = OB

Thus, ΔOAB is an equilateral triangle.

That means ∠AOB = ∠OBA = ∠OAB = 60°

Also, we know that the angle subtended by an arc at the centre of the circle is twice the angle subtended by it at any other point in the remaining part of the circle.

So, ∠AOB = 2∠ACB

⇒ ∠ACB = 1/2 (∠AOB)

⇒ ∠ACB = 1/2 (60°)

= 30°

Hence, the angle subtended by the given chord at a point in the major segment is 30°.

Example 2:

Two chords AB and AC of a circle subtend angles equal to 90º and 150º, respectively at the centre. Find ∠BAC, if AB and AC lie on the opposite sides of the centre.

Solution:

Given,

Two chords AB and AC of a circle subtend angles equal to 90º and 150º.

So, ∠AOC = 90º and∠AOB = 150º

In ΔAOB,

OA = OB (radius of the circle)

As we know, angles opposite to equal sides are equal.

So, ∠OBA = ∠OAB

According to the angle sum property of triangle theorem, the sum of all angles of a triangle = 180°

In ΔAOB,

∠OAB + ∠AOB +∠OBA = 180°

∠OAB + 90° + ∠OAB = 180°

2∠OAB = 180° – 90°

2∠OAB = 90°

⇒ ∠OAB = 45°

Now, in ΔAOC,

OA = OC (radius of the circle)

As mentioned above, angles opposite to equal sides are equal.

∴ ∠OCA = ∠OAC

Using the angle sum property in ΔAOB, we have;

∠OAC + ∠AOC +∠OCA = 180°

∠OAC + 150° + ∠OAC = 180°

2∠OAC = 180° – 150°

2∠OAC = 30°

⇒ ∠OAC = 15°

Now, ∠BAC = ∠OAB + ∠OAC

= 45° + 15°

= 60°

Therefore, ∠BAC = 60°

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?

A circle is defined as a closed two-dimensional figure whose all the points in the boundary are equidistant from a single point (called centre).

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.

The length of any chord can be calculated using the following formula:

Chord Length = 2 × √(r2 − d2)

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.