1. **State the problem:** We have a regular hexagon with side length 1 cm, and we need to find the value of $x$, which is the length of a vertical segment inside the hexagon. 2. **Recall properties of a regular hexagon:** A regular hexagon can be divided into 6 equilateral triangles, each with side length 1 cm. 3. **Find the height of one equilateral triangle:** The height $h$ of an equilateral triangle with side length $s$ is given by: $$h = \frac{\sqrt{3}}{2} s$$ Since $s=1$, $$h = \frac{\sqrt{3}}{2} \times 1 = \frac{\sqrt{3}}{2}$$ 4. **Relate $x$ to the height of the equilateral triangle:** The vertical segment $x$ corresponds to the height of two stacked equilateral triangles (since the hexagon height is twice the height of one triangle): $$x = 2 \times h = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}$$ 5. **Final answer:** $$x = \sqrt{3}$$ --- 1. **State the problem:** Prove that in a 45°-45°-90° triangle, the hypotenuse is $\sqrt{2}$ times as long as each leg. 2. **Recall the properties of a 45°-45°-90° triangle:** It is an isosceles right triangle with legs of equal length $a$ and hypotenuse $c$. 3. **Use the Pythagorean theorem:** $$c^2 = a^2 + a^2 = 2a^2$$ 4. **Solve for $c$:** $$c = \sqrt{2a^2} = a\sqrt{2}$$ 5. **Conclusion:** The hypotenuse is $\sqrt{2}$ times the length of each leg.