1. **Problem 4: Find the area of the L-shaped polygon with external dimensions 8 cm by 12 cm and an internal cut of 4 cm by 4 cm.** 2. The formula for the area of a rectangle is $\text{Area} = \text{length} \times \text{width}$. 3. Calculate the area of the large rectangle: $$12 \text{ cm} \times 8 \text{ cm} = 96 \text{ cm}^2$$ 4. Calculate the area of the cut-out small rectangle: $$4 \text{ cm} \times 4 \text{ cm} = 16 \text{ cm}^2$$ 5. Subtract the cut-out area from the large rectangle area to get the L-shape area: $$96 - 16 = 80 \text{ cm}^2$$ --- 6. **Problem 5: Verify the rectangle inside the circle with sides 3 in and 11 in and diagonal 7 in.** 7. Use the Pythagorean theorem for the rectangle: $$\text{diagonal}^2 = \text{length}^2 + \text{width}^2$$ 8. Calculate the sum of squares of sides: $$3^2 + 11^2 = 9 + 121 = 130$$ 9. Calculate the square of the diagonal: $$7^2 = 49$$ 10. Since $130 \neq 49$, the diagonal length 7 in is incorrect for a rectangle with sides 3 in and 11 in. --- 11. **Problem 6: Find the area of the cross-shaped polygon made of rectangles with arms measuring 5 m, 4 m, and 5 m in both horizontal and vertical directions.** 12. The cross shape can be seen as a large square minus four small corner squares. 13. Calculate the total length of one side of the large square: $$5 + 4 + 5 = 14 \text{ m}$$ 14. Area of the large square: $$14 \times 14 = 196 \text{ m}^2$$ 15. Each corner square has side length 5 m, so area of one corner square: $$5 \times 5 = 25 \text{ m}^2$$ 16. Total area of four corner squares: $$4 \times 25 = 100 \text{ m}^2$$ 17. Area of the cross-shaped polygon: $$196 - 100 = 96 \text{ m}^2$$