1. **Problem Statement:** Verify the properties of quadrilateral KLMN with vertices K(-1, 4), L(2, 2), M(0, -1), and N(-3, 1): a) Check if it is a square. b) Verify each diagonal is the perpendicular bisector of the other. c) Verify the diagonals are equal in length. 2. **Formulas and Rules:** - Distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$: $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ - Midpoint formula: $$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$ - Two lines are perpendicular if the product of their slopes is $-1$. - A square has four equal sides and four right angles. 3. **Calculate side lengths:** - $KL=\sqrt{(2-(-1))^2+(2-4)^2}=\sqrt{3^2+(-2)^2}=\sqrt{9+4}=\sqrt{13}$ - $LM=\sqrt{(0-2)^2+(-1-2)^2}=\sqrt{(-2)^2+(-3)^2}=\sqrt{4+9}=\sqrt{13}$ - $MN=\sqrt{(-3-0)^2+(1-(-1))^2}=\sqrt{(-3)^2+2^2}=\sqrt{9+4}=\sqrt{13}$ - $NK=\sqrt{(-1-(-3))^2+(4-1)^2}=\sqrt{2^2+3^2}=\sqrt{4+9}=\sqrt{13}$ All sides are equal length $\sqrt{13}$. 4. **Calculate slopes of sides to check right angles:** - $m_{KL}=\frac{2-4}{2-(-1)}=\frac{-2}{3}=-\frac{2}{3}$ - $m_{LM}=\frac{-1-2}{0-2}=\frac{-3}{-2}=\frac{3}{2}$ - $m_{MN}=\frac{1-(-1)}{-3-0}=\frac{2}{-3}=-\frac{2}{3}$ - $m_{NK}=\frac{4-1}{-1-(-3)}=\frac{3}{2}$ Check perpendicularity of adjacent sides: - $m_{KL} \times m_{LM} = -\frac{2}{3} \times \frac{3}{2} = -1$ - $m_{LM} \times m_{MN} = \frac{3}{2} \times -\frac{2}{3} = -1$ - $m_{MN} \times m_{NK} = -\frac{2}{3} \times \frac{3}{2} = -1$ - $m_{NK} \times m_{KL} = \frac{3}{2} \times -\frac{2}{3} = -1$ All adjacent sides are perpendicular, confirming right angles. 5. **Calculate diagonals:** - $KM=\sqrt{(0-(-1))^2+(-1-4)^2}=\sqrt{1^2+(-5)^2}=\sqrt{1+25}=\sqrt{26}$ - $LN=\sqrt{(-3-2)^2+(1-2)^2}=\sqrt{(-5)^2+(-1)^2}=\sqrt{25+1}=\sqrt{26}$ Diagonals are equal in length. 6. **Check if diagonals bisect each other:** - Midpoint of $KM$: $\left(\frac{-1+0}{2}, \frac{4+(-1)}{2}\right) = \left(-\frac{1}{2}, \frac{3}{2}\right)$ - Midpoint of $LN$: $\left(\frac{2+(-3)}{2}, \frac{2+1}{2}\right) = \left(-\frac{1}{2}, \frac{3}{2}\right)$ Midpoints are the same, so diagonals bisect each other. 7. **Check if diagonals are perpendicular:** - Slope of $KM$: $\frac{-1-4}{0-(-1)}=\frac{-5}{1}=-5$ - Slope of $LN$: $\frac{1-2}{-3-2}=\frac{-1}{-5}=\frac{1}{5}$ Product of slopes: $-5 \times \frac{1}{5} = -1$, so diagonals are perpendicular. **Final conclusion:** Quadrilateral KLMN has equal sides, right angles, equal and perpendicular diagonals that bisect each other, so it is a square.