Question

Three girls Ishita, Isha and Nisha are playing a game by standing on a circle of radius 20 m drawn in a park. Ishita throws a ball to Isha, Isha to Nisha and Nisha to Ishita. If the distance between Ishita and Isha and between Isha and Nisha is 24 m each, what is the distance between Ishita and Nisha.

Answer 1

Using the data given in the question we can draw a diagram that looks like fig (1).

From the figure we see that it is an isosceles triangle that has been circumscribed in a circle of radius R = 20 m.

The equal sides of the isosceles triangle measure 24 m in length. The length of the base of the isosceles triangle is what we are required to find out.

Since it is an isosceles triangle the perpendicular dropped from the vertex A to the base will pass though the circumcentre of the triangle. Let ‘h’ be the height of the triangle.

Since the triangle has been circumscribed by a circle of radius ‘R’ the length of the distances from ‘O’ to any of the three persons would be ‘R’.

Let the positions of the persons Isha, Ishita and Nisha be replaced by ‘A’, ‘B’ and ‘C’ respectively. And let the length of the unknown base be, BC = 2x m.

This is shown in the fig (2).

Now, consider the triangle ΔBOD, we have

At the same time consider, we have

Substitute this value in equation we got for ‘R’, we get

Now we have got the value of the height of the triangle as h = 14.4 m.

Substituting the value of h in the below equation,

Now we have the value of x = 19.2 m

We need the value of

Hence, the distance between Ishita and Nisha is .