1. **Problem Statement:** Find the values of $x$ and $y$ that make each quadrilateral a parallelogram. 2. **Key Property:** In a parallelogram, the diagonals bisect each other. This means the segments of the diagonals are equal in pairs. --- ### I. Find $x$ and $y$ for each quadrilateral **1. First quadrilateral:** - Given diagonal segments: $12$, $27$, $2x + 9$, and $3x + 6$ - Since diagonals bisect each other, set equal pairs: $$12 = 2x + 9$$ $$27 = 3x + 6$$ **Solve for $x$: ** 1. From $12 = 2x + 9$, subtract 9: $$12 - 9 = 2x \Rightarrow 3 = 2x$$ 2. Divide both sides by 2: $$x = \frac{3}{2} = 1.5$$ **Check with second equation:** $$27 = 3(1.5) + 6 = 4.5 + 6 = 10.5$$ This does not equal 27, so re-check the problem or assume only the first pair is used for $x$. **2. Second quadrilateral:** - Diagonal segments: $2x + 9$, $2x$, $3x - 7$, and $5y - 3$ - Set equal pairs: $$2x + 9 = 3x - 7$$ $$2x = 5y - 3$$ **Solve for $x$ and $y$: ** 1. From $2x + 9 = 3x - 7$, subtract $2x$ and add 7: $$9 + 7 = 3x - 2x \Rightarrow 16 = x$$ 2. Substitute $x=16$ into $2x = 5y - 3$: $$2(16) = 5y - 3 \Rightarrow 32 = 5y - 3$$ 3. Add 3 to both sides: $$35 = 5y$$ 4. Divide by 5: $$y = 7$$ --- ### II. Solve the following **a. Find the perimeter of the parallelogram given sides $a$ and $b$:** - Formula: $$P = 2(a + b)$$ 1. $a=12$, $b=4$: $$P = 2(12 + 4) = 2(16) = 32$$ 2. $a=23$, $b=11$: $$P = 2(23 + 11) = 2(34) = 68$$ 3. $a=31$, $b=15$: $$P = 2(31 + 15) = 2(46) = 92$$ **b. Find the area of the parallelogram given base $b$ and height $h$:** - Formula: $$A = b \times h$$ 1. $b=23$, $h=21$: $$A = 23 \times 21 = 483$$ 2. $b=13$, $h=11$: $$A = 13 \times 11 = 143$$ 3. $b=45$, $h=25$: $$A = 45 \times 25 = 1125$$ --- **Final answers:** I. 1. $x = 1.5$, $y$ not determined from given info 2. $x = 16$, $y = 7$ II. a. Perimeters: 32, 68, 92 b. Areas: 483, 143, 1125