1. **State the problem:** Ruby has 4 identical right-angled triangles with sides 12 cm, 16 cm, and 20 cm. She arranges them to form a star. We need to find the perimeter of the star. 2. **Identify the triangle type and sides:** The triangle sides are 12 cm, 16 cm, and 20 cm. Since $12^2 + 16^2 = 144 + 256 = 400$ and $20^2 = 400$, this confirms the triangle is right-angled with hypotenuse 20 cm. 3. **Understand the star shape:** The star is formed by placing 4 identical right triangles with their right angles at the center, so the star's perimeter consists of the outer edges of the triangles. 4. **Calculate the perimeter:** Each triangle contributes two sides to the star's perimeter (the legs 12 cm and 16 cm), but the hypotenuses (20 cm) meet inside the star and do not form the perimeter. 5. **Sum the perimeter:** The perimeter is the sum of the 4 triangles' legs on the outside: $$\text{Perimeter} = 4 \times (12 + 16) = 4 \times 28 = 112$$ 6. **Final answer:** The perimeter of the star is $112$ cm.