Equal Chords and Equal Angles

Class 9Circles

The relationship between equal chords and the angles they subtend at the centre of a circle is a fundamental theorem in the NCERT Class 9 Circles chapter.

This theorem establishes that equal chords subtend equal angles at the centre, and conversely, chords subtending equal angles at the centre are equal. A related result shows that equal chords are equidistant from the centre.

These theorems connect chord lengths, central angles, and distances from the centre, forming the backbone of circle geometry in Class 9.

What is Equal Chords and Equal Angles?

Theorem 1: Equal chords of a circle subtend equal angles at the centre.

If chord AB = chord CD, then ∠AOB = ∠COD (where O is the centre).

Theorem 2 (Converse): If two chords of a circle subtend equal angles at the centre, then the chords are equal.

If ∠AOB = ∠COD, then chord AB = chord CD.

Theorem 3: Equal chords of a circle are equidistant from the centre.

Theorem 4 (Converse): Chords equidistant from the centre of a circle are equal.

Key terminology:

Chord: A line segment joining any two points on the circle.
Central angle: The angle subtended by a chord at the centre of the circle.
Distance from centre to chord: The perpendicular distance from the centre to the chord.

Equal Chords and Equal Angles Formula

Key Results and Formulas:

1. Equal chords ⇒ Equal central angles:

AB = CD ⇒ ∠AOB = ∠COD

2. Equal central angles ⇒ Equal chords:

∠AOB = ∠COD ⇒ AB = CD

3. Relationship between chord length and distance from centre:

If the radius is r, the chord length is l, and the perpendicular distance from centre to chord is d:

d² + (l/2)² = r²

Where:

d = perpendicular distance from centre to chord
l = length of the chord
r = radius of the circle

4. Longer chord is closer to the centre:

If chord AB > chord CD, then the distance of AB from the centre is less than the distance of CD from the centre.
The diameter is the longest chord and passes through the centre (distance = 0).

Derivation and Proof

Proof: Equal chords subtend equal angles at the centre

Given: A circle with centre O. Chords AB = CD.

To prove: ∠AOB = ∠COD.

Proof:

In ΔAOB and ΔCOD:
OA = OC (radii of the same circle) ... (i)
OB = OD (radii of the same circle) ... (ii)
AB = CD (given) ... (iii)
By SSS congruence: ΔAOB ≅ ΔCOD.
Therefore, ∠AOB = ∠COD (CPCT).

Proof of Converse: Equal angles at centre ⇒ Equal chords

Given: ∠AOB = ∠COD.

To prove: AB = CD.

In ΔAOB and ΔCOD:
OA = OC (radii) ... (i)
OB = OD (radii) ... (ii)
∠AOB = ∠COD (given) ... (iii)
By SAS congruence: ΔAOB ≅ ΔCOD.
Therefore, AB = CD (CPCT).

Proof: Equal chords are equidistant from centre

Given: AB = CD. OM ⊥ AB and ON ⊥ CD (M, N are feet of perpendiculars).

To prove: OM = ON.

The perpendicular from the centre bisects the chord.
AM = AB/2 and CN = CD/2. Since AB = CD, AM = CN. ... (i)
In ΔOMA and ΔONC:
OA = OC (radii)
AM = CN (from (i))
∠OMA = ∠ONC = 90°
By RHS congruence: ΔOMA ≅ ΔONC.
Therefore, OM = ON (CPCT).

Types and Properties

Important Observations and Corollaries:

1. Diameter is the longest chord:

The diameter passes through the centre, so its distance from the centre is 0.
Any other chord is at a positive distance from the centre and is therefore shorter.

2. Chord length increases as distance decreases:

As a chord moves closer to the centre, it gets longer.
As it moves away from the centre, it gets shorter.

3. Congruent circles:

Equal chords of congruent circles (circles with the same radius) subtend equal angles at their respective centres.
All four theorems extend naturally to congruent circles.

4. Central angle and arc:

Equal chords cut equal arcs (minor arcs are equal, major arcs are equal).
The central angle equals the arc's degree measure.

5. Perpendicular bisector passes through centre:

The perpendicular bisector of any chord always passes through the centre of the circle.
This is the basis for finding the centre when given a chord.

Solved Examples

Example 1: Example 1: Finding the central angle

Problem: In a circle with centre O, chord AB = chord CD = 8 cm. If ∠AOB = 60°, find ∠COD.

Solution:

Given:

AB = CD = 8 cm
∠AOB = 60°

By the theorem: Equal chords subtend equal angles at the centre.

∠COD = ∠AOB = 60°

Answer: ∠COD = 60°.

Example 2: Example 2: Finding chord length from equal angles

Problem: In a circle with centre O, ∠AOB = ∠COD = 45°. If AB = 10 cm, find CD.

Solution:

Given:

∠AOB = ∠COD = 45°
AB = 10 cm

By the converse theorem: Chords subtending equal angles at the centre are equal.

CD = AB = 10 cm

Answer: CD = 10 cm.

Example 3: Example 3: Distance from centre to chord

Problem: A chord of length 24 cm is drawn in a circle of radius 13 cm. Find the distance of the chord from the centre.

Solution:

Given:

Chord length l = 24 cm, Radius r = 13 cm

Using d² + (l/2)² = r²:

d² + 12² = 13²
d² + 144 = 169
d² = 25
d = 5 cm

Answer: The distance from the centre = 5 cm.

Example 4: Example 4: Equal chords equidistant from centre

Problem: Two chords AB and CD of a circle are each 16 cm long. The radius is 10 cm. Find the distance of each chord from the centre and verify they are equal.

Solution:

For chord AB (l = 16 cm, r = 10 cm):

d₁² + 8² = 10²
d₁² = 100 − 64 = 36
d₁ = 6 cm

For chord CD (l = 16 cm, r = 10 cm):

d₂² + 8² = 10²
d₂ = 6 cm

Answer: Both chords are at 6 cm from the centre. Equal chords are equidistant from the centre. ✓

Example 5: Example 5: Finding chord length given distance

Problem: Two chords are equidistant from the centre of a circle of radius 15 cm. Each chord is at a distance of 9 cm from the centre. Find the length of each chord.

Solution:

Using d² + (l/2)² = r²:

9² + (l/2)² = 15²
81 + (l/2)² = 225
(l/2)² = 144
l/2 = 12
l = 24 cm

By the converse theorem: Chords equidistant from the centre are equal.

Answer: Each chord is 24 cm long.

Example 6: Example 6: Which chord is longer?

Problem: In a circle of radius 17 cm, chord PQ is at a distance of 8 cm from the centre and chord RS is at a distance of 15 cm. Which chord is longer?

Solution:

Chord PQ:

(l₁/2)² = 17² − 8² = 289 − 64 = 225
l₁/2 = 15, so l₁ = 30 cm

Chord RS:

(l₂/2)² = 17² − 15² = 289 − 225 = 64
l₂/2 = 8, so l₂ = 16 cm

Answer: PQ (30 cm) is longer than RS (16 cm). The chord closer to the centre is longer.

Example 7: Example 7: Prove equal arcs correspond to equal chords

Problem: In a circle with centre O, arc AB = arc CD. Prove that chord AB = chord CD.

Solution:

Equal arcs subtend equal angles at the centre.
If arc AB = arc CD, then ∠AOB = ∠COD.
By the converse of the equal chords theorem: if ∠AOB = ∠COD, then chord AB = chord CD.

Hence proved: Equal arcs have equal chords.

Example 8: Example 8: Two circles with equal chords

Problem: Two congruent circles have centres O₁ and O₂. Chord AB in circle 1 equals chord CD in circle 2. If ∠AO₁B = 80°, find ∠CO₂D.

Solution:

The circles are congruent (same radius).
AB = CD (given).
Equal chords of congruent circles subtend equal angles at their respective centres.
∠CO₂D = ∠AO₁B = 80°

Answer: ∠CO₂D = 80°.

Example 9: Example 9: Three equal chords

Problem: Three chords AB, CD, and EF of a circle are equal. If they are at distances d₁, d₂, d₃ from the centre, find the relationship between these distances.

Solution:

By the theorem: Equal chords are equidistant from the centre.

AB = CD = EF ⇒ d₁ = d₂ = d₃

Also:

∠AOB = ∠COD = ∠EOF (equal chords subtend equal central angles).

Answer: d₁ = d₂ = d₃. All three chords are equidistant from the centre.

Example 10: Example 10: Finding radius from equal chord data

Problem: Two equal chords of a circle, each of length 48 cm, are at a distance of 7 cm from the centre. Find the radius of the circle.

Solution:

Given:

Chord length l = 48 cm, distance d = 7 cm

Using d² + (l/2)² = r²:

7² + 24² = r²
49 + 576 = r²
r² = 625
r = 25 cm

Answer: The radius is 25 cm.

Real-World Applications

Applications of Equal Chords and Equal Angles:

Circle construction: Dividing a circle into equal arcs (for regular polygons) relies on equal chords subtending equal angles.
Gear design: Equal teeth on gears correspond to equal chords and equal central angles, ensuring smooth rotation.
Architecture: Circular arches and domes use equal chord spacing for symmetrical construction.
Clock design: The 12 hour markings on a clock face are endpoints of equal chords, each subtending 30° at the centre.
Navigation: Circular radar displays use equal chord properties for distance and angle calculations.
Proving other theorems: The equal chords theorem is used to prove results about cyclic quadrilaterals, inscribed angles, and circle symmetry.

Key Points to Remember

Equal chords of a circle subtend equal angles at the centre (proved by SSS congruence).
Converse: Chords subtending equal angles at the centre are equal (proved by SAS congruence).
Equal chords are equidistant from the centre (proved by RHS congruence).
Converse: Chords equidistant from the centre are equal.
The relationship d² + (l/2)² = r² connects chord length, distance from centre, and radius.
The longer a chord, the closer it is to the centre.
The diameter is the longest chord (distance from centre = 0).
Equal chords cut equal arcs.
All four theorems extend to congruent circles (circles with equal radii).
This topic is part of Chapter 10 (Circles) in CBSE Class 9 NCERT Mathematics.

Practice Problems

Two chords AB and CD of a circle with centre O are equal. If ∠AOB = 70°, find ∠COD.
A chord of length 30 cm is at a distance of 8 cm from the centre. Find the radius.
Two chords PQ and RS are equidistant from the centre. PQ = 20 cm. Find RS.
In a circle of radius 25 cm, find the distance of a chord of length 48 cm from the centre.
Two chords AB = 12 cm and CD = 16 cm are in a circle of radius 10 cm. Which chord is closer to the centre?
Prove that equal chords of congruent circles are equidistant from their respective centres.
Three equal chords are drawn in a circle. Prove that the triangle formed by joining their midpoints is equilateral.
A chord is at a distance of 5 cm from the centre of a circle of radius 13 cm. Find the chord length and the angle it subtends at the centre.

Frequently Asked Questions

Q1. What is the relationship between equal chords and central angles?

Equal chords subtend equal angles at the centre. Conversely, chords that subtend equal angles at the centre are equal in length. This is proved using triangle congruence (SSS and SAS).

Q2. What does 'equidistant from the centre' mean?

It means the perpendicular distance from the centre of the circle to each chord is the same. The perpendicular from the centre to a chord bisects the chord.

Q3. Is the diameter the longest chord?

Yes. The diameter passes through the centre, so its perpendicular distance from the centre is 0. Since chord length decreases as distance from centre increases, the diameter is the longest chord.

Q4. How do you find the distance of a chord from the centre?

Use the formula d² + (l/2)² = r², where d is the perpendicular distance, l is the chord length, and r is the radius. Solve for d.

Q5. Do equal chords always cut equal arcs?

Yes. Equal chords subtend equal central angles, and the arc length is determined by the central angle. So equal chords cut equal minor arcs and equal major arcs.

Q6. Can two unequal chords be equidistant from the centre?

No. By the theorem, chords equidistant from the centre must be equal. Unequal chords are at different distances from the centre.

Q7. Does this apply to chords in different circles?

Yes, provided the circles are congruent (have equal radii). Equal chords of congruent circles subtend equal angles at their respective centres and are equidistant from their respective centres.

Q8. Is this topic in CBSE Class 9?

Yes. Equal chords and their properties are covered in Chapter 10 (Circles) of the CBSE Class 9 NCERT Mathematics textbook.

Equal Chords and Equal Angles

What is Equal Chords and Equal Angles?

Equal Chords and Equal Angles Formula

Derivation and Proof

Types and Properties

Solved Examples

Example 1: Example 1: Finding the central angle

Example 2: Example 2: Finding chord length from equal angles

Example 3: Example 3: Distance from centre to chord

Example 4: Example 4: Equal chords equidistant from centre

Example 5: Example 5: Finding chord length given distance

Example 6: Example 6: Which chord is longer?

Example 7: Example 7: Prove equal arcs correspond to equal chords

Example 8: Example 8: Two circles with equal chords

Example 9: Example 9: Three equal chords

Example 10: Example 10: Finding radius from equal chord data

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is the relationship between equal chords and central angles?

Q2. What does 'equidistant from the centre' mean?

Q3. Is the diameter the longest chord?

Q4. How do you find the distance of a chord from the centre?

Q5. Do equal chords always cut equal arcs?

Q6. Can two unequal chords be equidistant from the centre?

Q7. Does this apply to chords in different circles?

Q8. Is this topic in CBSE Class 9?

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