Chord Properties of a Circle

This section covers fundamental properties of chords in circles, essential for understanding circle geometry. We will explore how chords relate to the center of the circle, the angles they subtend, and their distances from the center. Understanding these properties allows us to solve geometric problems involving circles and their related elements.

Formulae

• Angle Subtended by a Chord: The angle subtended by a chord at the center of the circle is twice the angle subtended by it at any point on the major arc.
• Perpendicular from Center: A perpendicular drawn from the center of a circle to a chord bisects the chord.
• Distance and Equality: Equal chords in a circle are equidistant from the center.

Examples

Let’s illustrate these concepts with a couple of examples:

Example-1: In a circle, chord AB and chord CD are equal in length. If the distance of AB from the center O is 5 cm, what is the distance of CD from the center O?

Solution: Since equal chords are equidistant from the center, the distance of CD from O is also 5 cm.

Example-2: In a circle, a perpendicular from the center O to chord PQ intersects PQ at point M. If PM = 8 cm, what is the length of chord PQ?

Solution: The perpendicular from the center bisects the chord, so MQ = PM = 8 cm. Therefore, PQ = PM + MQ = 8 cm + 8 cm = 16 cm.

Theorem with Proof

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

Given: A circle with center O. Chords AB and CD are equal in length (AB = CD).

To Prove: $ \angle AOB = \angle COD $

Proof:

1. Consider triangles $ \triangle AOB $ and $ \triangle COD $.
2. $ OA = OC $ (Radii of the same circle)
3. $ OB = OD $ (Radii of the same circle)
4. $ AB = CD $ (Given)
5. Therefore, $ \triangle AOB \cong \triangle COD $ (By SSS congruence criterion).
6. Hence, $ \angle AOB = \angle COD $ (By CPCTC – Corresponding Parts of Congruent Triangles).

Common Mistakes by Students

• Confusing Radius and Chord: Students sometimes mix up the concepts of radius and chord, leading to incorrect application of properties.
• Incorrect Application of Converse: Failing to recognize the converse of a theorem, such as assuming that if two angles are equal, the chords are equal without proper justification.
• Ignoring Perpendicularity: Incorrectly applying the perpendicular bisector property, particularly when the perpendicular is not explicitly stated or drawn.
• Using Incorrect Congruence Criteria: Difficulty in identifying the correct congruence criterion (SSS, SAS, ASA) to prove triangle congruence.

Real Life Application

Chord properties are fundamental in various real-life applications:

• Engineering: Used in the design of circular structures such as bridges, tunnels, and arches.
• Architecture: Applied in creating circular features, understanding the stability of buildings, and calculating areas related to circles.
• Navigation: The properties of circles are crucial for calculating distances and angles in GPS systems and other navigational tools.
• Manufacturing: Used for the production of circular objects such as gears, wheels, and circular-shaped machine components.

Fun Fact

The word “chord” comes from the Latin word “chorda”, which means “bowstring”. Ancient mathematicians used to measure the lengths of chords to calculate angles and distances in circles, much like measuring the length of the string of a bow!

Recommended YouTube Videos for Deeper Understanding

Q.1 In a circle, two chords $AB$ and $CD$ are equal. If $\angle AOB = 60^\circ$, where $O$ is the center, then what is the measure of $\angle COD$?

Check Solution

Ans: B

Equal chords subtend equal angles at the center. Therefore, $\angle COD = \angle AOB = 60^\circ$.

Q.2 A chord of length 8 cm is drawn in a circle of radius 5 cm. What is the distance of the chord from the center of the circle?

Check Solution

Ans: B

The perpendicular from the center bisects the chord. Consider a right-angled triangle formed by the radius, half the chord, and the distance from the center. Using the Pythagorean theorem: $5^2 = 4^2 + d^2$. So, $d = \sqrt{25-16} = 3$ cm.

Q.3 If a line segment drawn from the center of a circle to a chord bisects the chord, then what is the relationship between the line segment and the chord?

Check Solution

Ans: B

The perpendicular from the center to a chord bisects the chord (and converse). Therefore, the line segment is perpendicular to the chord.

Q.4 Two chords, $PQ$ and $RS$, of a circle are equidistant from the center. Which of the following statements is true?

Check Solution

Ans: C

Equal chords are equidistant from the center (and converse). Since the chords are equidistant, they must be equal in length.

Q.5 How many circles can be drawn through three non-collinear points?

Check Solution

Ans: B

Through three non-collinear points, one and only one circle can be drawn.

Next Topic: Angle Properties of a Circle (Class 9)