To find the perimeter of the triangle, we need to find the lengths of its sides.
Let's start by finding the radius of the circle. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. From the given information, we have:
A = 25Ï€ cm^2
Setting this equal to the formula, we get:
25π = πr^2
Dividing both sides by π, we have:
25 = r^2
Taking the square root of both sides, we get:
r = √25
r = 5 cm
Since the diameter AB coincides with one of the sides of the triangle ACB, the length of AB is equal to the diameter of the circle, which is 2 * radius = 2 * 5 = 10 cm.
Now, let's find the height of the triangle. Since vertex C lies on the circle, the height of the triangle is equal to the radius of the circle, which is 5 cm.
The area of the triangle is given by the formula A = (1/2) * base * height. We can rearrange this formula to solve for the base:
base = (2 * A) / height
Plugging in the values, we get:
base = (2 * 11) / 5
base = 22 / 5
base = 4.4 cm
Now, we can use the Pythagorean theorem to find the length of the remaining side of the triangle. Let's call it side AC. Here's the setup:
AC^2 = AB^2 - BC^2
Substituting the values, we get:
AC^2 = 10^2 - 4.4^2
AC^2 = 100 - 19.36
AC^2 = 80.64
Taking the square root of both sides, we get:
AC = √80.64
AC ≈ 8.971 cm
Finally, we can find the perimeter of the triangle by adding up the lengths of all its sides:
Perimeter = AB + BC + AC
Perimeter = 10 + 4.4 + 8.971
Perimeter ≈ 23.371 cm
Therefore, the perimeter of the triangle is approximately 23.371 cm.