Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5 mdrawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If thedistance between Reshma and Salma and between Salma and Mandip is 6 m each, what is thedistance between Reshma and Mandip

The problem can be solved using the properties of an equilateral triangle and the Pythagorean theorem.

Step 1: Understand the problem The three girls are standing on a circle, and the distance between Reshma and Salma and between Salma and Mandip is 6 m each. This means that Reshma, Salma, and Mandip form an equilateral triangle (all sides are equal) inside the circle. We need to find the distance between Reshma and Mandip.

Step 2: Use the properties of an equilateral triangle In an equilateral triangle, all angles are 60 degrees. Therefore, the angle between Reshma, Salma, and Mandip is 60 degrees.

Step 3: Use the Pythagorean theorem The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can apply this theorem to the triangle formed by Reshma, Salma, and Mandip.

Let's denote:

• the distance between Reshma and Salma as a = 6 m,
• the distance between Salma and Mandip as b = 6 m,
• the distance between Reshma and Mandip as c (this is what we need to find).

Since the triangle is equilateral, it can be split into two right triangles by drawing a line from Salma (the top of the triangle) perpendicular to the base (the line between Reshma and Mandip). This line splits the base into two equal parts, each of length c/2.

In each of these right triangles, the hypotenuse is a (or b), one side is c/2, and the other side is the height of the equilateral triangle, which we'll denote as h.

According to the Pythagorean theorem, we have:

a^2 = (c/2)^2 + h^2, which simplifies to: 36 = c^2/4 + h^2.

We also know that in an equilateral triangle, the height h can be found using the formula h = sqrt(3)/2 * a, which in this case gives h = sqrt(3)/2 * 6 = 3sqrt(3).

Substituting h = 3sqrt(3) into the equation, we get:

36 = c^2/4 + 9*3, which simplifies to: 36 = c^2/4 + 27.

Subtracting 27 from both sides gives:

9 = c^2/4.

Multiplying both sides by 4 gives:

36 = c^2.

Finally, taking the square root of both sides gives:

c = 6 m.

So, the distance between Reshma and Mandip is also 6 m.