1. **Problem Statement:** We have two right-angled triangles joined by a common side. One triangle has an angle of 60° and a vertical side labeled $n$. The other triangle has angles 30° and 60°, with the hypotenuse labeled $x$. We need to find $x$ in terms of $n$. 2. **Recall the properties of 30°-60°-90° triangles:** - The sides are in the ratio $1 : \sqrt{3} : 2$. - The side opposite 30° is the shortest side. - The side opposite 60° is $\sqrt{3}$ times the shortest side. - The hypotenuse is twice the shortest side. 3. **Analyze the first triangle (with angle 60° and vertical side $n$):** - The vertical side $n$ is opposite the 60° angle. - Let the shortest side (opposite 30°) be $s$. - Then $n = s \sqrt{3}$, so $s = \frac{n}{\sqrt{3}}$. 4. **Analyze the second triangle (with hypotenuse $x$ and angles 30° and 60°):** - The hypotenuse $x$ is twice the shortest side of this triangle. - The shortest side here is the same as the common side shared with the first triangle, which is $s = \frac{n}{\sqrt{3}}$. 5. **Find $x$ in terms of $n$:** - Since $x$ is twice the shortest side, we have $$x = 2s = 2 \times \frac{n}{\sqrt{3}} = \frac{2n}{\sqrt{3}}.$$ 6. **Final answer:** $$\boxed{x = \frac{2n}{\sqrt{3}}}.$$