Question

In a circle of radius \[5\ cm\], calculate the length of the chord which is at the distance of \[3\ cm\] from the centre.

Answer 1

Hint: In this question, we need to find the length of the chord which is at the distance of \[3\ cm\] from the centre. Also given that the radius of the circle is \[5\ cm\].First we can draw the figure according to the given . Then, we can apply the properties of the circle to obtain a right triangle. Then we can use the Pythagoras theorem in the right triangle formed . By using this we can find the length of a chord which is at a distance of \[3\ cm\] from the centre of a circle .

Complete step by step answer:
Given that the radius of the circle is \[5\] cm. First we can draw the figure according to the given .

Let OA be the radius of the radius which is given as\[\ 5\ cm\]. Here OC is the perpendicular distance from centre O to the chord AB and is of \[3\] cm. Also here the perpendicular drawn from the centre bisects the chord.Then AC will be equal to CB. Let us consider AC as \[x\]. Then the length of the chord will be \[2x\]. Since AOC forms a right angle triangle, we can apply Pythagoras theorem. The Pythagoras theorem is
\[\left( \text{Hypotenuse} \right)^{2} = \left( \text{Perpendicular} \right)^{2} + \ \left( \text{base} \right)^{2}\]
That is \[\left(AO \right)^{2} = \left(OC \right)^{2} + \left(AC \right)^{2}\]

Now on substituting the known values,
We get,
\[\left( 5 \right)^{2} = \left( 3 \right)^{2} + \left(AC \right)^{2}\]
On simplifying,
We get,
\[\Rightarrow \ 25 = 9 + \left(AC \right)^{2}\]
On subtracting both sides by \[9\] ,
We get,
\[\Rightarrow \ 25 – 9 = \left( AC \right)^{2}\]
On simplifying,
We get,
\[\Rightarrow AC^{2} = 16\]

On taking square root on both sides,
We get,
\[\Rightarrow AC = \pm \sqrt{16}\]
On simplifying,
We get,
\[\therefore AC = \pm 4\]
Thus \[AC = 4\ cm\] (since the length cannot be negative).Now we can calculate the length of chord AB which is twice of AC.That is \[AB = 8\ cm\].Thus we get the length of the chord is \[8\ cm\] .

Therefore, the length of the chord is \[8\ cm\].

Note: In order to solve these types of questions, we should have a strong grip over the properties of the circle. The chord is nothing but line segments whose endpoints both lie on the circle . We also need to know that mathematically, in a circle when a line a drawn from the centre to any point on the circle then that line is known as the radius of the circle similarly when a line is drawn from a end point to the other end point of the circle passing through the centre of the circle is known as diameter of a circle.