A chord of length 40 cm is at a distance of 15 cm from the centre. What is the length of the chord of the same circle which is at a distance of 20 cm from the centre of circle?

Given:

A circle with a chord of length 40 cm which is 15 cm away from the centre of the circle.

We are to find the length of another chord in the same circle which is 20 cm away from the centre.

Concept Used:

The Perpendicular from the centre of a circle to a chord bisects the chord.

Pythagorean theorem: Hypotenuse²  = Base² + Perpendicular²

Calculation:

MP = PN = MN/2 = 40/2 = 20 cm, and

The distance from the centre as d1 = 15 cm.

From the Pythagorean theorem, the radius r of the circle is:

⇒ \(r = \sqrt(a^2 + d1^2)\) 

⇒ \(\sqrt((20 cm)^2 + (15 cm)^2)\)

⇒ \(\sqrt(400 cm^2 + 225 cm^2)\)

⇒ \(\sqrt(625 cm^2)\)

⇒ 25 cm.

For the unknown chord, let's denote half of its length as x/2, and the known distance from the centre as d2 = 20 cm.

Again using the Pythagorean theorem, we find b as:

 \(x/2 = \sqrt(r^2 - d2^2)\)

⇒ \(\sqrt((25 cm)^2 - (20 cm)^2)\)

⇒ \(\sqrt(625 cm^2 - 400 cm^2)\)

⇒ \(\sqrt(225 cm^2)\)

⇒ 15 cm.

So, the full length of the unknown chord is x = 2 × 15 cm = 30 cm.

Therefore, the length of the chord of the same circle which is at a distance of 20 cm from the center of the circle is 30 cm.

∴ Option 4 is the correct answer.