1. **Problem statement:** A rectangular piece of cardboard has length 210 mm and width 297 mm. We need to find the areas of the largest isosceles triangle and the largest right triangle that can be cut from the cardboard. 2. **Formula for area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Important rules:** - The largest triangle inscribed in a rectangle will have its base equal to one side of the rectangle and height equal to the other side. - For an isosceles triangle, the base is one side of the rectangle and the height is the other side. - For a right triangle, the legs are the sides of the rectangle. 4. **Largest isosceles triangle:** - Base = 297 mm - Height = 210 mm - Area = $$\frac{1}{2} \times 297 \times 210 = \frac{1}{2} \times 62370 = 31185 \text{ mm}^2$$ 5. **Largest right triangle:** - The right triangle with legs equal to the length and width of the rectangle. - Base = 210 mm - Height = 297 mm - Area = $$\frac{1}{2} \times 210 \times 297 = 31185 \text{ mm}^2$$ 6. **Conclusion:** Both the largest isosceles and right triangles have the same area of 31185 mm². Final answer: - Area of largest isosceles triangle = 31185 mm² - Area of largest right triangle = 31185 mm²