Step-by-step explanation:

Sure, let's solve these step by step:

1. In a right-angled triangle ABC, you can use the Pythagorean Theorem to find the length of the hypotenuse (c) when the lengths of the other two sides (a and b) are known. The Pythagorean Theorem states that:

c² = a² + b²

(i) For the first case:

a = 2.5 m, b = 6 m

c² = 2.5² + 6²

c² = 6.25 + 36

c² = 42.25

c = √42.25

c ≈ 6.5 m

(ii) For the second case:

a = 7.5 m, b = 18 m

c² = 7.5² + 18²

c² = 56.25 + 324

c² = 380.25

c = √380.25

c ≈ 19.5 m

2. In this case, we have AC = 5 cm and BC = 12 cm, and the angle is right-angled at C. You can use the Pythagorean Theorem again to find the length of the hypotenuse AB:

AB² = AC² + BC²

AB² = 5² + 12²

AB² = 25 + 144

AB² = 169

AB = √169

AB = 13 cm

So, the length of AB is 13 cm.