Question

parallelogram wxyz has vertices at w (1, 3), x (2, 5), y (6, 3), and z (5, 1). determine whether wxyz is a rectangle, rhombus, or square.
select all the true statements.
show hints
square
rectangle
rhombus

Explanation:

Step1: Calculate side lengths

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

• $WX=\sqrt{(2-1)^2+(5-3)^2}=\sqrt{1+4}=\sqrt{5}$
• $XY=\sqrt{(6-2)^2+(3-5)^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$
• $YZ=\sqrt{(5-6)^2+(1-3)^2}=\sqrt{1+4}=\sqrt{5}$
• $ZW=\sqrt{(1-5)^2+(3-1)^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$

Step2: Check for right angles

Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$

• Slope of $WX$: $m_{WX}=\frac{5-3}{2-1}=2$
• Slope of $XY$: $m_{XY}=\frac{3-5}{6-2}=\frac{-2}{4}=-\frac{1}{2}$
• Product: $2\times(-\frac{1}{2})=-1$, so $WX\perp XY$

Step3: Classify the parallelogram

Opposite sides equal, angles are right angles.

Answer:

rectangle