1. **Problem Statement:** Find the values of $x$ and $y$ that make each quadrilateral a parallelogram based on the given diagonal segment lengths. 2. **Important Property:** In a parallelogram, the diagonals bisect each other. This means each diagonal is divided into two equal parts by their intersection. --- ### First Quadrilateral: - Given diagonal segments: 12, 27, $2y + 9$, and $3x + 6$. - Since diagonals bisect each other, opposite segments are equal: - $12 = 2y + 9$ - $27 = 3x + 6$ **Step 1:** Solve for $y$ from $12 = 2y + 9$: $$12 = 2y + 9$$ $$12 - 9 = 2y$$ $$3 = 2y$$ $$y = \frac{3}{2} = 1.5$$ **Step 2:** Solve for $x$ from $27 = 3x + 6$: $$27 = 3x + 6$$ $$27 - 6 = 3x$$ $$21 = 3x$$ $$x = \frac{21}{3} = 7$$ --- ### Second Quadrilateral: - Given diagonal segments: $2y + 9$, $2x$, $3x - 7$, and $5y - 3$. - Diagonals bisect each other, so: - $2y + 9 = 3x - 7$ - $2x = 5y - 3$ **Step 3:** Solve the system: From $2y + 9 = 3x - 7$: $$2y + 9 = 3x - 7$$ $$2y - 3x = -16$$ From $2x = 5y - 3$: $$2x - 5y = -3$$ Rewrite system: $$2y - 3x = -16$$ $$2x - 5y = -3$$ Multiply second equation by $1.5$ to align $x$ terms: $$3x - 7.5y = -4.5$$ Add to first equation: $$(2y - 3x) + (3x - 7.5y) = -16 - 4.5$$ $$2y - 7.5y = -20.5$$ $$-5.5y = -20.5$$ $$y = \frac{-20.5}{-5.5} = \frac{20.5}{5.5} = \frac{41}{11} \approx 3.727$$ Substitute $y$ into $2x - 5y = -3$: $$2x - 5 \times \frac{41}{11} = -3$$ $$2x - \frac{205}{11} = -3$$ $$2x = -3 + \frac{205}{11} = -\frac{33}{11} + \frac{205}{11} = \frac{172}{11}$$ $$x = \frac{172}{22} = \frac{86}{11} \approx 7.818$$ --- ### Final answers for part I: - First quadrilateral: $x = 7$, $y = 1.5$ - Second quadrilateral: $x = \frac{86}{11}$, $y = \frac{41}{11}$ --- **Slug:** parallelogram variables **Subject:** geometry **Desmos:** {"latex":"","features":{"intercepts":true,"extrema":true}} **q_count:** 2