1. **Problem statement:** We have two right triangles sharing a side. The larger triangle has a side of 10 cm adjacent to a 45° angle. The smaller triangle has a 60° angle and an opposite side labeled $x$ cm. We want to find the length $x$. 2. **Step 1: Analyze the larger triangle.** - The larger triangle has a 45° angle and a side adjacent to it of length 10 cm. - In a right triangle with a 45° angle, the sides adjacent and opposite to the angle are equal because it is an isosceles right triangle. - Therefore, the side opposite the 45° angle is also 10 cm. 3. **Step 2: Analyze the smaller triangle.** - The smaller triangle has a 60° angle and shares the side opposite the 45° angle of the larger triangle, which is 10 cm. - This side (10 cm) is adjacent to the 60° angle in the smaller triangle. 4. **Step 3: Use trigonometry to find $x$.** - In the smaller triangle, $x$ is the side opposite the 60° angle. - Using the tangent function: $$\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{10}$$ 5. **Step 4: Calculate $x$.** - We know $\tan(60^\circ) = \sqrt{3}$. - So, $$\sqrt{3} = \frac{x}{10} \implies x = 10 \times \sqrt{3}$$ 6. **Final answer:** $$x = 10\sqrt{3} \approx 17.32 \text{ cm}$$ This means the length $x$ is approximately 17.32 cm.