In a 30-60-90 triangle, the side opposite the 30° angle (let's call it "a") is half the length of the hypotenuse (let's call it "c"), and the side opposite the 60° angle (let's call it "b") is √3 times the length of the side opposite the 30° angle:

a = c/2
b = a√3

Now we know that the hypotenuse has a length of 16 cm:

a = 16/2
a = 8 cm

Now we can find the length of side b:

b = 8√3 cm

Since ΔABC is a right triangle with ∠BCA = 60° and AB = 8, we can find the length of the hypotenuse AC using the Pythagorean theorem:AC² = AB² + BC²
AC² = 8² + (8 * tan(60°))²
AC² = 64 + 64 * (√3)²
AC² = 64 + 64 * 3
AC² = 64 + 192
AC² = 256Now, find the length of the hypotenuse AC:AC = √256 = 16 cmSince the hypotenuse serves as the diameter of the circle, the radius R is half the length of the hypotenuse:R = AC / 2 = 16 / 2 = 8 cmFinally, find the area of the circle using the formula:Area = π * R²In this case, the area of the circle is:Area = π * (8)² = 64π square centimeters

Since the triangle is a right triangle with one angle of 60°, the other acute angle must be 30°. The triangle is a 30-60-90 triangle. In a 30-60-90 triangle, the side adjacent to the 60° angle (let's call it "a") is half the length of the hypotenuse (let's call it "c").

In this case, the hypotenuse has a length of 17 cm. Therefore, the length of side a (adjacent to angle 60°) is 17/2 = 8.5 cm.

Since the triangle is a right triangle with one angle of 30°, the other acute angle must be 60°. The triangle is a 30-60-90 triangle. In a 30-60-90 triangle, the side opposite the 30° angle (let's call it "a") is half the length of the hypotenuse (let's call it "c"):

a = c/2

Now we know that the hypotenuse has a length of 8 cm:

a = 8/2
a = 4 cm

In a 30-60-90 triangle, the hypotenuse (c) is twice the length of the side opposite the 30° angle (a). The side opposite the 60° angle (b) is √3 times the length of the side opposite the 30° angle:

b = a√3

We are given the side length opposite the 60° angle as 14 cm:

14 cm = a√3

Divide both sides by √3:

a = 14/√3 cm

Now we can find the length of the hypotenuse:

c = 2a = 2(14/√3) cm = 28/√3 cm