Radius of a Circle and Chord: Radius of a Circle and Solved Example

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Radius of a Circle

A circle can be defined as locus of a point moving in a plane, in such a manner that its distance from a fixed point is always constant. The fixed point is known as the center of the circle and distance between any point on the circle and its center is called the radius of a circle.

Given a line and a Circle, it could either be touching the circle or non-touching.

Consider any line AB and a circle. Then according to the relative positions of the line and the circle, three possibilities can arise as shown in the given figure.

Radius of a circle

Radius of a Circle

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Line AB intersects the given circle at two distinct points P and Q. The line AB in this case is referred to as secant of the circle. Points P and Q lie on the circumference of the circle, but they do not pass through center of the circle ‘O’, hence line segment PQ is known as a chord of the circle as its endpoints lie on the circle.

Therefore, chord of a circle can be defined as a line segment joining any two distinct points on circle’s circumference. A chord passing through the center of circle is known as diameter of the circle and it is the largest chord of the circle. This diameter is twice that of the radius of a circle i.e. , where ‘D’ is the diameter and is the radius.

Radius of a circle

Radius of a Circle

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Radius of a circle

Or,

Diameter of a circle

Let us discuss few important theorems and their proofs related to chord of circle.

Theorem 1: The perpendicular line drawn from the center of a circle to a chord bisects the chord.

Radius of a circle

Radius of a Circle

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Given:

To prove:

Construction: Draw

Proof:

The perpendicular line drawn from the center of a circle to a chord bisects the chord.

Sr No.

Statement

Reason

In

1

Radii of the same circle

2

Common

3

Each angle measure

4

By RHS congruence criterion

5

By CPCT (Corresponding parts of congruent triangles)

The converse of the above theorem is also true.

Theorem 2: The line drawn through the center of the circle to bisect a chord is perpendicular to the chord.

Radius of a circle

Radius of a Circle

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Given: C is the midpoint of the chord AB of the circle with center of circle at O

To prove:

Construction: Join

Proof:

C is the midpoint of the chord AB of the circle with center of circle at O

Sr. No

Statement

Reason

1.

Radii of the same circle

2.

Common

3.

Given

4.

SSS Axion of congruency

5.

Corresponding parts of congruent triangle

6.

Linear pair Axion

7.

From statement and

8.

Following the above statements

Solved Example

Let us see some solved problems on radius and chord of a circle.

Example: Find the radius of the circle if its diameter is

Solution:

Given,

Diameter of circle

We have formula to find the radius

Put the value of diameter,

Example 2: If the length of the chord of a circle is 12 cm and the perpendicular distance from the center to the chord is 8 cm, then what is the radius of the circle?

Solution:

Let us draw a circle as per the given information.

Radius of a circle

Radius of a Circle

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Length of the chord

Perpendicular distance

Radius

We know that the, the perpendicular line drawn from the center of a circle to a chord bisects the chord.

In triangle OPA,

By Pythagoras theorem,

Put the value,

Therefore, radius of the circle is 10 cm.