1. **State the problem:** We have a compound L-shaped figure with some side lengths given and one unknown vertical side length. We need to find the unknown side length (a) and then calculate the perimeter (b). 2. **Identify known sides:** - Left vertical side: 15 cm - Top horizontal side: 4 cm - Bottom horizontal side: 9 cm - Inner horizontal segment: 5 cm - Lower right vertical side: 8 cm - Unknown vertical side: ? cm 3. **Find the unknown side length:** The total vertical length on the left is 15 cm. The vertical sides on the right are split into two parts: the unknown side and the 8 cm side. Since the shape is compound and the vertical sides must add up to the same total height, we have: $$15 = ? + 8$$ 4. **Solve for the unknown side:** $$? = 15 - 8$$ $$? = 7$$ 5. **Calculate the perimeter:** The perimeter is the sum of all outer side lengths. The horizontal sides are 4 cm (top), 5 cm (inner horizontal), and 9 cm (bottom). The vertical sides are 15 cm (left), 7 cm (unknown), and 8 cm (lower right). Sum horizontal sides: $$4 + 5 + 9 = 18$$ Sum vertical sides: $$15 + 7 + 8 = 30$$ Total perimeter: $$18 + 30 = 48$$ **Final answers:** - Unknown side length: $7$ cm - Perimeter: $48$ cm