In a 45-45-90 triangle, the hypotenuse is √2 times the length of each leg (sides opposite the 45° angles). We are given the length of one leg (opposite a 45° angle) as 8 cm. To find the length of the hypotenuse (let's call it "c"), we can use the following equation:

c = 8√2 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 triangle JKL is a right triangle with angle K = 60°, angle L must be 30° (because angle J is a right angle). The triangle JKL is, therefore, a 30-60-90 triangle. In a 30-60-90 triangle, the hypotenuse (JL) is twice the length of the side opposite the 30° angle (KL), and the side JK (opposite the 60° angle) has a length that is √3 times the side opposite the 30° angle (KL).

Let "x" be the length of side KL, then JL is 2x and JK is x√3.

Perimeter = KL + JK + JL
30 cm = x + x√3 + 2x

Now we can solve for x:

x(1 + √3 + 2) = 30 cm
x ≈ 5 cm

The length of side KL is approximately 5 cm.

Since triangle LMN is a right triangle with angle M = 45°, angle N must be 45° as well (because angle L is a right angle). The triangle LMN is, therefore, a 45-45-90 triangle. In a 45-45-90 triangle, the sides opposite both of the 45° angles are congruent, and the hypotenuse is √2 times the length of each leg.

In this case, the side LN (opposite angle N) is 9 cm. Since the triangle is a 45-45-90 triangle, the side opposite angle M (MN) must also be 9 cm.

A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the Pythagorean theorem, a² + b² = c².
First, we will rationalize the first option. (3,4,5): (3 cm)² + (4 cm)² = (5 cm)²?; 9 + 16 = 25; 25 = 25.
Next, (5,12,13): (5 cm)² + (12 cm)² = (13 cm)²?; 25 + 144 = 169; 169 = 169.
Now for (7,24,25): (7 cm)² + (24 cm)² = (25 cm)²?; 49 + 576 = 625; 625 = 625.
Last, (8,15,18): (8 cm)² + (15 cm)² = (18 cm)²?; 64 + 225 = 289; 289 ≠ 324.

The Pythagorean triple that doesn't meet the condition is (8, 15, 18).