1. **Problem statement:** Find the perimeter of each figure given the side lengths. 2. **Formula for perimeter:** The perimeter $P$ of a polygon is the sum of the lengths of all its sides. 3. **Important rules:** - For rectangles and parallelograms, opposite sides are equal. - Add all side lengths carefully, combining like terms. --- ### a) Rectangle with sides $2x - 1$ and $3r^2 + 10$ 1. Opposite sides are equal, so perimeter: $$P = 2 \times (2x - 1) + 2 \times (3r^2 + 10)$$ 2. Multiply out: $$P = 2(2x - 1) + 2(3r^2 + 10) = 4x - 2 + 6r^2 + 20$$ 3. Combine like terms: $$P = 4x + 6r^2 + 18$$ --- ### b) Irregular polygon with sides $x$, $-5x - 13$, $-5x^2 - y$, and $9x^2 + y + 1$ 1. Sum all sides: $$P = x + (-5x - 13) + (-5x^2 - y) + (9x^2 + y + 1)$$ 2. Combine like terms carefully: $$P = (x - 5x) + (-13) + (-5x^2 + 9x^2) + (-y + y) + 1$$ 3. Simplify: $$P = -4x - 13 + 4x^2 + 0 + 1 = 4x^2 - 4x - 12$$ --- ### c) Triangle with sides $r^2$, $5r^2 - 12$, and $3r^2 - 8r$ 1. Sum all sides: $$P = r^2 + (5r^2 - 12) + (3r^2 - 8r)$$ 2. Combine like terms: $$P = r^2 + 5r^2 - 12 + 3r^2 - 8r = (r^2 + 5r^2 + 3r^2) - 8r - 12$$ 3. Simplify: $$P = 9r^2 - 8r - 12$$ --- ### d) Parallelogram with opposite sides $2w^2 - w + 3$ and $w^2 - \frac{1}{2}w$ 1. Opposite sides are equal, so perimeter: $$P = 2 \times (2w^2 - w + 3) + 2 \times \left(w^2 - \frac{1}{2}w\right)$$ 2. Multiply out: $$P = 2(2w^2 - w + 3) + 2\left(w^2 - \frac{1}{2}w\right) = 4w^2 - 2w + 6 + 2w^2 - w$$ 3. Combine like terms: $$P = (4w^2 + 2w^2) + (-2w - w) + 6 = 6w^2 - 3w + 6$$ --- **Final answers:** - a) $P = 4x + 6r^2 + 18$ - b) $P = 4x^2 - 4x - 12$ - c) $P = 9r^2 - 8r - 12$ - d) $P = 6w^2 - 3w + 6$