1. **Stating the problem:** We have an irregular L-shaped polygon with given side lengths: bottom horizontal length 32 cm, left vertical height 22 cm, top left horizontal segment 8 cm, and right vertical segment 15 cm. We need to find the perimeter $M$ and area $N$ of this figure, then calculate $M + N$. 2. **Understanding the shape:** The polygon is L-shaped with right angles. To find the perimeter, we sum all outer side lengths. To find the area, we can split the L-shape into two rectangles and sum their areas. 3. **Finding missing side lengths:** - The total vertical height on the left is 22 cm. - The right vertical segment is 15 cm. - The vertical segment between these two is $22 - 15 = 7$ cm. - The bottom horizontal length is 32 cm. - The top left horizontal segment is 8 cm. - The horizontal segment between these two is $32 - 8 = 24$ cm. 4. **Perimeter calculation:** The perimeter $M$ is the sum of all outer sides: $$M = 22 + 8 + 15 + 24 + 7 + 32$$ Calculate step-by-step: $$22 + 8 = 30$$ $$30 + 15 = 45$$ $$45 + 24 = 69$$ $$69 + 7 = 76$$ $$76 + 32 = 108$$ So, $M = 108$ cm. 5. **Area calculation:** Split the L-shape into two rectangles: - Rectangle 1 (left vertical part): width = 8 cm, height = 22 cm - Rectangle 2 (bottom horizontal part): width = 24 cm, height = 15 cm Calculate areas: $$\text{Area}_1 = 8 \times 22 = 176$$ $$\text{Area}_2 = 24 \times 15 = 360$$ Total area $N$: $$N = 176 + 360 = 536$$ cm² 6. **Final result:** $$M + N = 108 + 536 = 644$$ **Answer:** $M + N = 644$