1. **State the problem:** We have an L-shaped polygon with given side lengths 4 cm, 6 cm, and 10 cm, and an unknown top horizontal side labeled as [Box] cm. The total perimeter is 38 cm. 2. **Identify the sides:** The polygon has the following sides: - Left vertical side: 4 cm - Inner vertical side: 6 cm - Right vertical side: 10 cm - Bottom horizontal side: unknown, but can be found using the perimeter - Top horizontal side: [Box] cm (unknown) 3. **Use the perimeter formula:** The perimeter $P$ is the sum of all side lengths. 4. **Find the missing horizontal sides:** Since the shape is L-shaped, the bottom horizontal side plus the top horizontal side equals the total horizontal length. The vertical sides are 4 cm, 6 cm, and 10 cm. 5. **Calculate the sum of vertical sides:** $$4 + 6 + 10 = 20$$ 6. **Calculate the sum of horizontal sides:** $$\text{Perimeter} - \text{sum of vertical sides} = 38 - 20 = 18$$ 7. **Find the bottom horizontal side:** The bottom horizontal side is the same length as the top horizontal side plus the horizontal segment adjacent to the 6 cm vertical side. Since the problem only gives one unknown horizontal side, we assume the bottom horizontal side is equal to the top horizontal side plus the horizontal segment adjacent to the 6 cm vertical side. 8. **Set up the equation:** Let $x$ be the length in the box (top horizontal side). Then the bottom horizontal side is $x + 6$ (since the horizontal segment adjacent to the 6 cm vertical side is 6 cm). 9. **Sum of horizontal sides:** $$x + (x + 6) = 18$$ 10. **Solve for $x$:** $$2x + 6 = 18$$ $$2x = 18 - 6$$ $$2x = 12$$ $$x = \frac{12}{2}$$ $$x = 6$$ 11. **Answer:** The number in the box is 6 cm.