1. **State the problem:** We have a right triangle with legs 8 cm and 15 cm. Inside the triangle on the hypotenuse, two squares are drawn, one with an area of 9 cm², and the other with an unknown area. We want to find the unknown square's area. 2. **Analyze given information:** - Smaller square area = 9 cm², so side length $s_1 = \sqrt{9} = 3$ cm. - The triangle legs are 8 cm and 15 cm, so the hypotenuse length is calculated by Pythagoras: $$ c = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \text{ cm} $$ 3. **Geometry insight:** The two squares are on the hypotenuse, divided into two segments of equal length (marked by equal segments on the hypotenuse), so each segment is half the hypotenuse: $$ \frac{17}{2} = 8.5 \text{ cm} $$ 4. **Relation of squares to these segments:** The sides of the squares correspond to the lengths of segments on the hypotenuse where the squares are drawn. The smaller square side is 3 cm, so it must be on a segment of the hypotenuse of length 3 cm, but the mark indicates two equal segments of length 8.5 cm. Instead, the smaller square must correspond to one of the legs or other segments within the triangle. 5. **Alternative approach using the triangle similarity:** Consider the smaller square of side 3 cm inside the right triangle. Since the triangle's legs are 8 and 15 cm, and squares are drawn on the legs as per the figure logic, we check the squares on legs. 6. **Check if the smaller square is on leg 8 cm:** The square with area 9 cm² has side 3 cm, so cannot fully cover the leg 8 cm. 7. **Check if the smaller square is on segment formed on the hypotenuse:** Since the hypotenuse is split into two equal segments, each is 8.5 cm, and the smaller square side is 3 cm, so the smaller square is within one segment, and larger square within the other. 8. **Find the larger square side length:** Since side of smaller square is 3 cm, assume the smaller and larger squares' side lengths sum up to 8.5 cm: $$ s_1 + s_2 = 8.5 $$ $$ s_2 = 8.5 - 3 = 5.5 \text{ cm} $$ 9. **Calculate the larger square's area:** $$ \text{Area} = s_2^2 = 5.5^2 = 30.25 \text{ cm}^2 $$ **Final answer:** The unknown square's area is $30.25$ cm².