1. **Problem Statement:** Given parallelogram PURE with diagonals PR and EU intersecting at O. We know diagonals of a parallelogram bisect each other, so $O$ is the midpoint of both $PR$ and $EU$. **Given:** - $PO = 7x - 20$ cm - $PR = 4x + 10$ cm Find: a. $x$ b. $\frac{PR}{PO}$ c. $\frac{PO}{PR}$ 2. **Step 1: Use the midpoint property of diagonals** Since $O$ is midpoint of $PR$, $PO = OR = \frac{PR}{2}$. So, $$PO = \frac{PR}{2}$$ Substitute given expressions: $$7x - 20 = \frac{4x + 10}{2}$$ 3. **Step 2: Solve for $x$** Multiply both sides by 2: $$2(7x - 20) = 4x + 10$$ $$14x - 40 = 4x + 10$$ Bring terms to one side: $$14x - 4x = 10 + 40$$ $$10x = 50$$ Divide both sides by 10: $$\cancel{10}x = \frac{50}{\cancel{10}}$$ $$x = 5$$ 4. **Step 3: Calculate $PR$ and $PO$ using $x=5$** $$PO = 7(5) - 20 = 35 - 20 = 15 \text{ cm}$$ $$PR = 4(5) + 10 = 20 + 10 = 30 \text{ cm}$$ 5. **Step 4: Calculate ratios** $$\frac{PR}{PO} = \frac{30}{15} = 2$$ $$\frac{PO}{PR} = \frac{15}{30} = \frac{1}{2}$$ **Final answers:** a. $x = 5$ b. $\frac{PR}{PO} = 2$ c. $\frac{PO}{PR} = \frac{1}{2}$