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The radius of a circle is 13 cm and the length of one of its chords is 10 cm. The distance of the chord from the centre is-
A. 8 cm
B. 10 cm
C. 12 cm
D. 6 cm

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: Here, we have to find the distance of the chord from the centre. We will find the chord length using Perpendicular distance from the centre. The chord of a circle can be defined as the line segment joining any two points on the circumference of the circle.

 

Formula Used:

Chord length Using Perpendicular distance from the centre is given by \[{\text{Chord Length}} = 2 \times \sqrt {\left( {{r^2} - {d^2}} \right)} \], where \[d\] is the perpendicular distance of the chord from the centre and \[r\] is the radius of the circle.

 

Complete Step by Step Solution:

We will draw a circle with radius of a circle \[r\] as 13 cm and chord of a circle \[l\] as 10 cm.

 

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Let AC be the chord of a circle, OP be the radius of the circle and OB be the distance between the chord of a circle and radius of a circle.

Length of the chord \[ = 10{\text{cm}}\]

\[ \Rightarrow l = \dfrac{{10}}{2}\]

Dividing the terms, we get

\[ \Rightarrow l = {\text{5 cm}}\]

By using the formula \[{\text{Chord Length}} = 2 \times \sqrt {\left( {{r^2} - {d^2}} \right)} \], we have

\[d = \sqrt {{r^2} - {{\left( {\dfrac{l}{2}} \right)}^2}} \]

Substituting the values \[r = {\text{13 cm}}\] and \[\dfrac{l}{2} = {\text{5 cm}}\]

\[ \Rightarrow d = \sqrt {{{13}^2} - {5^2}} \]

The square of the number 13 is 169.

The square of the number 5 is 25.

Squaring the terms, we get

\[ \Rightarrow d = \sqrt {169 - 25} \]

\[ \Rightarrow d = \sqrt {144} \]

Computing the square root, we get

\[ \Rightarrow d = {\text{12 cm}}\]

 

Therefore, the distance of the chord from the centre is 12cm.

 

Note:

We know that in a circle, perpendicular from the centre bisects the chord. Among the properties of the chord of a circle, Chords are equidistant from the center if and only if their lengths are equal. A chord that passes through the center of a circle is called a diameter and is the longest chord. A radius or diameter that is perpendicular to a chord divides the chord into two equal parts. It means that both the halves of the chords are equal in length. The perpendicular bisector of a chord passes through the center of a circle. The formula can also be used to find the length of the cord and the radius of the circle by rewriting the equation.