Three girls, Ishita Isha and Nisha are plyaing a game by standing on a circle of radius 20 m drawan 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.
To find the distance between Ishita and Nisha, we can use the Law of Cosines. Let's label the circle's center as O and the points where Ishita, Isha, and Nisha are standing as A, B, and C, respectively.
We know that the radius of the circle is 20 m, so the distance between the center O and each of the points A, B, and C is 20 m.
Next, let's label the angles formed at the center of the circle O as α, β, and γ, with the respective arc lengths being a, b, and c.
We are given the distance between Ishita and Isha (segment AB) as 24 m and the distance between Isha and Nisha (segment BC) as 24 m. Therefore, the length of segment AC is what we need to find.
Using the Law of Cosines, we have the formula:
c^2 = a^2 + b^2 - 2ab * cos(γ)
In this case, a = 24 m, b = 24 m, and γ = 120 degrees (since angle γ is an equalateral triangle angle inscribed in a circle).
Plugging the values into the formula, we have:
c^2 = 24^2 + 24^2 - 2 * 24 * 24 * cos(120)
Now, let's calculate the value of cos(120):
cos(120) = -0.5
Substituting the value into the formula:
c^2 = 24^2 + 24^2 - 2 * 24 * 24 * (-0.5)
Simplifying:
c^2 = 576 + 576 + 576
c^2 = 1728
Taking the square root of both sides to find c:
c = √1728
c ≈ 41.57
Therefore, the distance between Ishita and Nisha (segment AC) is approximately 41.57 meters.