To find the lengths of the sides of the cyclic quadrilateral, we need to use the properties of a cyclic quadrilateral and the given information.
Let's first define some variables:
Let the two diagonals of the cyclic quadrilateral be AC and BD, where AC coincides with the diameter of the circle.
Let the intersection point of the diagonals be E.
From the given information, we know that the area of the circle is 36π cm². The formula for the area of a circle is given by A = πr², where A is the area and r is the radius. Since the area is 36π cm², we can equate this to πr² and solve for r:
36π = πr²
Dividing both sides by π, we get:
36 = r²
Taking the square root of both sides, we find:
r = 6
Since AC coincides with the diameter, its length is 2 times the radius, which is 2 * 6 = 12 cm.
We are also given that BD is the other diagonal and that it measures 8 cm. Additionally, we know that AC and BD intersect at right angles, so we have a right-angled triangle ADE.
Using the Pythagorean theorem, we can find the length of the third side DE. According to the theorem, in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, AC is the hypotenuse, so its length is 12 cm. The length of BD is 8 cm, and we need to find DE. Applying the Pythagorean theorem, we have:
12² = DE² + 8²
144 = DE² + 64
Subtracting 64 from both sides, we get:
80 = DE²
Taking the square root of both sides, we find:
DE = √80
Simplifying the square root, we have:
DE = 4√5 cm
Now, we have the lengths of two sides of the cyclic quadrilateral: AC = 12 cm and BD = 8 cm. To find the lengths of the other two sides, we need to use the property of a cyclic quadrilateral that opposite sides are supplementary.
Since AC and BD intersect at right angles, the quadrilateral is a rectangle. In a rectangle, opposite sides are equal in length.
Therefore, the lengths of the other two sides of the cyclic quadrilateral are also 12 cm and 8 cm.
In summary, the lengths of the sides of the cyclic quadrilateral are:
AC = 12 cm
BD = 8 cm
DE = 4√5 cm
The other two sides are also 12 cm and 8 cm.