1. **Problem 1: What additional information would prove that LMNP is a rectangle?** Given LMNP is a parallelogram, to prove it is a rectangle, we need to show that one angle is a right angle (90 degrees). 2. **Key fact:** A parallelogram is a rectangle if and only if one pair of adjacent sides are perpendicular. 3. **Options analysis:** - Lengths alone (√45 and √5) do not guarantee right angles. - Slope of LP and MN being -2 means those sides are parallel but does not prove right angles. - LM // PN is true for parallelograms but does not prove rectangle. - LP ⊥ PN means LP is perpendicular to PN, which proves a right angle. **Answer:** The statement "LP ⊥ PN" proves LMNP is a rectangle. 4. **Problem 2: How to find the perimeter of a quadrilateral given vertices?** 5. **Key formula:** Use the distance formula between two points $A(x_1,y_1)$ and $B(x_2,y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 6. **Steps:** - Calculate the length of each side using the distance formula. - Add all side lengths to get the perimeter. 7. **Answer:** Use the distance formula to find the length of each side, and then add the lengths.