In a circle of radius 10cm, A chord is drawn so that it is 4cm distant from the centre. Calculate the length of the minor arc cut off by the chord.

Answer 1

To find the length of the minor arc cut off by the chord, we need to find the angle subtended by that chord at the center of the circle.

Let's denote the center of the circle as O and the end points of the chord as P and Q. The chord PQ cuts off two arcs on the circle, a major arc and a minor arc. We are asked to find the length of the minor arc.

Since O is the center of the circle and OP and OQ are radii, triangle OPQ is an isosceles triangle with OP = OQ = 10 cm.

Also, the perpendicular distance from the center O to chord PQ is given as 4 cm. Let's denote it as OM.

In the right triangle OMP, using the Pythagorean Theorem, we can find the length of the half chord MP:

MP = √(OP² - OM²) = √(10² - 4²) = √(100 - 16) = √84 cm.

So the whole length of the chord PQ = 2 * MP = 2 * √84 cm.

In triangle OMP, sin(∠OMP) = OM / OP = 4 / 10 = 0.4

So, ∠OMP = arcsin(0.4) = 23.58 degrees.

Because ∠OMP is half the angle subtended by the chord at the center, the whole angle is 2 * ∠OMP = 2 * 23.58 = 47.16 degrees.

Because the ratio of the length of an arc to the circumference of the whole circle is equal to the ratio of the angle it subtends at the center to 360 degrees:

Length of the minor arc = (47.16 / 360) * Circumference
= (47.16 / 360) * 2πr
= (47.16 / 360) * 2π * 10
= 26.13 cm.

This is the length of the minor arc cut off by the chord.