Can the diameter be called a chord of the circle?

Yes. The diameter is a special chord passing through the centre.

Can two different chords have the same length?

Yes, they can. Typically, when they do, they are equidistant from the centre.

What is the maximum number of chords that can be drawn from one point?

The maximum number of chords that can be drawn from a single point inside a circle is infinite.

What is the chord of a circle formula with an angle?

The chord of a circle formula with angle is 2r sin (/2) where r = the radius of the circle; θ = the central angle in radians.

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Chord of Circle

Q: What is the chord of circle formula using the Pythagorean theorem?

Using the Pythagorean theorem, the chord length is 2√(r^2-d^2) Here, r represents the radius and d represents the perpendicular distance from the centre to the chord.

Understanding the concept of the chord of a circle is essential in geometry. Whether you are in high school or you are preparing for a competitive examination, understanding this fundamental concept is crucial for success. In this study guide, we will delve deep into the circle of chord definition, understand the formula, look into the circle of chord theorem, and illustrate with a circle of chord diagram. We will also include plenty of solved examples on chord of circle to improve your understanding.

1.0What is a Chord of a Circle?

Let’s begin with the chord of circle definition. It refers to a straight line segment that connects 2 points on the circumference of the circle. It is the longest distance between any two points of the circle. Chords are important in geometry as they are used to define circle properties like diameter or radius. A chord effectively divides the circle into two arcs. Below is a chord of a circle diagram for a better understanding.

Properties

A diameter is the longest chord of any given circle.
Any chord other than the diameter lies within the circle and does not pass through the centre.

2.0Chord of Circle Properties

Understanding the chord of circle properties solidifies the conceptual clarity. Refer to the table below to understand the properties in detail:

Properties	Detail
Equal Chords and Distance	From the centre, equal chords lie at equal distances.
Division of the Circle	A chord divides the circle into two segments: a minor segment and a major segment.
Secant	When a chord is extended infinitely on both sides, it becomes a secant (a line that intersects the circle at two points).
Diameter	A chord that passes through the circle’s centre is called a diameter and is the longest chord of that circle.
Isosceles Triangle	If you connect the ends of a chord to the centre of the circle, you form an isosceles triangle.

3.0Chord of Circle Theorems

Let’s go through some key Chord of Circle theorems that are used frequently in problem solving.

Perpendicular from the centre bisects the chord: If a perpendicular is drawn from the centre of a given circle to its chord, it bisects the chord.
Equal chords are equidistant from the centre: If two chords in any given circle are equal in length, then they are equidistant from the centre.
Equal angles subtend equal chords: If two chords subtend equal angles at the circle's centre, then the chords are equal in length.

4.0Chord of Circle Formula

The chord of circle formula helps us calculate the length of chords based on the information given in a mathematical problem.

Using Radius and Central Angle

The chord of circle formula with angle is:

Chord Length = $2 r \times s in (θ /2)$

Here,

r = the radius of the circle; θ = central angle in radians.

Using Pythagoras Theorem (When the perpendicular from the centre to the chord is known)

Chord Length: $2 \times r^{2} - d^{2}$

Here, r represents the radius and d represents the perpendicular distance from the centre to the chord.

5.0Applications of Chords in Geometry

There are many applications of chords of circles in real life that are very important.

Architecture and construction (arches, domes)
Astronomy (circular orbits)
Engineering (circular components and designs)
Geometry proofs and constructions

6.0Solved Examples on Chord of a Circle

Here are solved examples on chord of circle to put all the theory into practice:

Q 1: Find the length of a chord that subtends a 60° angle at the centre of a circle with a radius of 10 cm.

We know that Chord Length = $2 r \times s in (θ /2) = 2 \times 10 \times s in 30° = 20 \times 0.5 = 10 c m$

Q 2: The radius of a circle is 13 cm. A chord is 5 cm away from the centre. Find the length of the chord.

The formula for chord length is: $2 \times r^{2} - d^{2}$

Using it, we get = $2 \times 1 3^{2} - 5^{2} = 2 \times 169 - 25 = 2 \times 144 = 24 c m$

Q 3: Prove that the diameter is the longest chord of a circle of radius 7 cm.

Diameter = 2 × radius = 14 cm.

Any other chord will have length <2r. Hence, the diameter is the longest chord.

Q 4: Two chords AB and CD of a circle are each 12 cm long. If AB is 5 cm from the centre, find the distance from the centre to CD.

Since AB = CD and they are equal chords, they are equidistant. Distance to CD = 5 cm.

Q 5: Two chords subtend angles of 40° at the centre. Are they equal?

Yes. Equal angles at the centre imply equal chords.

Q 6: A chord of length 16 cm is at a distance of 6 cm from the centre. Find the radius.

Half chord is 8 cm.

$r = 8^{2} + 6^{2} = 64 + 36 = 100 = 10 c m$

7.0Conclusion

A chord of a circle is a geometric concept that is filled with properties, theorems, and real-world relevance. We have covered everything related to the chord of the circle in detail throughout the study guide. The visual learning and solved examples will help improve your understanding.