1. **State the problem:** We are given a right triangle PQR with a right angle at Q. Side PQ is vertical with length 8 cm, and the area of the triangle is 36 cm². We need to find the length of the hypotenuse PR. 2. **Formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Identify base and height:** Since angle Q is right, sides PQ and QR are perpendicular. We know PQ = 8 cm (height), and QR is the base (unknown length). 4. **Use the area to find QR:** $$36 = \frac{1}{2} \times QR \times 8$$ Multiply both sides by 2: $$72 = QR \times 8$$ Divide both sides by 8: $$QR = \frac{72}{8}$$ Show cancellation: $$QR = \frac{\cancel{72}}{\cancel{8}} = 9$$ So, QR = 9 cm. 5. **Find hypotenuse PR using Pythagoras theorem:** $$PR^2 = PQ^2 + QR^2$$ $$PR^2 = 8^2 + 9^2 = 64 + 81 = 145$$ $$PR = \sqrt{145}$$ 6. **Final answer:** $$PR = \sqrt{145} \approx 12.04 \text{ cm}$$