1. **State the problem:** We need to find expressions for the perimeter ($P$) and area ($A$) of three polygons with right angles, given side lengths involving $x$. 2. **Shape a:** - Sides: $x$, $x-2$, $x-3$, $x-4$ - The shape is an L-shaped polygon. To find perimeter, sum all outer sides. 3. **Perimeter of shape a:** $$P_a = x + (x-2) + (x-3) + (x-4)$$ Simplify: $$P_a = x + x - 2 + x - 3 + x - 4 = 4x - 9$$ 4. **Area of shape a:** Break the L-shape into two rectangles: - Rectangle 1: sides $x$ and $x-4$ - Rectangle 2: sides $(x-3)$ and $(x-2)$ Calculate areas: $$A_1 = x(x-4) = x^2 - 4x$$ $$A_2 = (x-3)(x-2) = x^2 - 5x + 6$$ Total area: $$A_a = A_1 - A_2 = (x^2 - 4x) - (x^2 - 5x + 6) = x - 6$$ 5. **Shape b:** - Sides: $1$, $x-1$, $x$, $2$ - Perimeter is sum of all sides: $$P_b = 1 + (x-1) + x + 2$$ Simplify: $$P_b = 1 + x - 1 + x + 2 = 2x + 2$$ 6. **Area of shape b:** Break into two rectangles: - Rectangle 1: $1$ by $x$ - Rectangle 2: $2$ by $(x-1)$ Calculate areas: $$A_1 = 1 \times x = x$$ $$A_2 = 2 \times (x-1) = 2x - 2$$ Total area: $$A_b = A_1 + A_2 = x + 2x - 2 = 3x - 2$$ 7. **Shape c:** - Sides: $3$, $x$, $x$, $4$ - Perimeter: $$P_c = 3 + x + x + 4 = 2x + 7$$ 8. **Area of shape c:** Break into two rectangles: - Rectangle 1: $3$ by $x$ - Rectangle 2: $4$ by $x$ Calculate areas: $$A_1 = 3x$$ $$A_2 = 4x$$ Total area: $$A_c = A_1 + A_2 = 3x + 4x = 7x$$ **Final answers:** - Shape a: $P = 4x - 9$, $A = x - 6$ - Shape b: $P = 2x + 2$, $A = 3x - 2$ - Shape c: $P = 2x + 7$, $A = 7x$