a) Determine the length of side AC. Step 1: Identify the given sides of the right-angled triangle ABC. The hypotenuse is BC = 25 cm. One leg is AB = 7 cm. The other leg is AC. Step 2: Apply the Pythagorean theorem. For a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: $$AC^2 + AB^2 = BC^2$$ Step 3: Substitute the known values into the equation. $$AC^2 + (7\,\text{cm})^2 = (25\,\text{cm})^2$$ $$AC^2 + 49\,\text{cm}^2 = 625\,\text{cm}^2$$ Step 4: Solve for $AC^2$. $$AC^2 = 625\,\text{cm}^2 - 49\,\text{cm}^2$$ $$AC^2 = 576\,\text{cm}^2$$ Step 5: Calculate AC by taking the square root. $$AC = \sqrt{576\,\text{cm}^2}$$ $$AC = 24\,\text{cm}$$ The length of side AC is $\boxed{24\,\text{cm}}$. b) Determine the area of the design. Step 1: Identify the base and height of the triangle. For a right-angled triangle, the legs can be considered the base and height. Base (AB) = 7 cm Height (AC) = 24 cm (from part a) Step 2: Use the formula for the area of a triangle. $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ Step 3: Substitute the values and calculate the area. $$\text{Area} = \frac{1}{2} \times 7\,\text{cm} \times 24\,\text{cm}$$ $$\text{Area} = \frac{1}{2} \times 168\,\text{cm}^2$$ $$\text{Area} = 84\,\text{cm}^2$$ The area of the design is $\boxed{84\,\text{cm}^2}$.