1. **Stating the problem:** We need to find the area and perimeter of three triangles: ABC, PQR, and DEF. Each triangle has two side lengths given, but the third side is missing. 2. **Important note:** To find the perimeter, we need all three side lengths. To find the area, we need either the height or all three sides to use Heron's formula. Since the third side is missing, we cannot find exact values without additional information. 3. **Assuming the triangles are right triangles:** If the two given sides are perpendicular, the third side can be found using the Pythagorean theorem: $$c = \sqrt{a^2 + b^2}$$ where $a$ and $b$ are the given sides, and $c$ is the hypotenuse. 4. **Calculations for each triangle:** **a) Triangle ABC:** Given sides: $5$ cm and $6$ cm Calculate third side: $$c = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} \approx 7.81$$ cm Perimeter: $$P = 5 + 6 + 7.81 = 18.81$$ cm Area (right triangle area formula): $$A = \frac{1}{2} \times 5 \times 6 = 15$$ cm$^2$ **b) Triangle PQR:** Given sides: $10$ cm and $8$ cm Calculate third side: $$c = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164} \approx 12.81$$ cm Perimeter: $$P = 10 + 8 + 12.81 = 30.81$$ cm Area: $$A = \frac{1}{2} \times 10 \times 8 = 40$$ cm$^2$ **c) Triangle DEF:** Given sides: $8$ cm and $10$ cm Calculate third side: $$c = \sqrt{8^2 + 10^2} = \sqrt{64 + 100} = \sqrt{164} \approx 12.81$$ cm Perimeter: $$P = 8 + 10 + 12.81 = 30.81$$ cm Area: $$A = \frac{1}{2} \times 8 \times 10 = 40$$ cm$^2$ 5. **Summary:** Assuming right triangles with the given sides as legs, we found the third side using the Pythagorean theorem, then calculated perimeter and area accordingly. If the triangles are not right triangles, more information is needed to solve the problem.