Question

1. it is given that wxyz is a parallelogram. plot the points and graph the figure. prove what type of parallelogram you have and justify your answer by using theorems about quadrilaterals. remember to show all steps.

w(7, -1), x(3, 6), y(-1, -1), z(3, -8)

Explanation:

Step1: Recall Parallelogram Theorems

A parallelogram with perpendicular diagonals is a rhombus. A parallelogram with congruent diagonals is a rectangle. First, find the diagonals of \(WXYZ\): diagonals are \(WY\) and \(XZ\).

Step2: Calculate Length of Diagonal \(WY\)

Points \(W(7, -1)\) and \(Y(-1, -1)\). Use distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
\(x_1 = 7, y_1 = -1\); \(x_2 = -1, y_2 = -1\).
\(d_{WY} = \sqrt{(-1 - 7)^2 + (-1 - (-1))^2} = \sqrt{(-8)^2 + 0^2} = \sqrt{64} = 8\).

Step3: Calculate Length of Diagonal \(XZ\)

Points \(X(3, 6)\) and \(Z(3, -8)\).
\(x_1 = 3, y_1 = 6\); \(x_2 = 3, y_2 = -8\).
\(d_{XZ} = \sqrt{(3 - 3)^2 + (-8 - 6)^2} = \sqrt{0^2 + (-14)^2} = \sqrt{196} = 14\). Wait, no—wait, check slopes for perpendicularity.

Step4: Calculate Slopes of Diagonals

Slope of \(WY\): \(m_{WY} = \frac{-1 - (-1)}{-1 - 7} = \frac{0}{-8} = 0\) (horizontal line).
Slope of \(XZ\): \(m_{XZ} = \frac{-8 - 6}{3 - 3}\), which is undefined (vertical line).

Step5: Determine Perpendicularity

A horizontal line (slope 0) and vertical line (undefined slope) are perpendicular (they intersect at 90°). Since diagonals of parallelogram \(WXYZ\) are perpendicular, by the theorem "If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus", \(WXYZ\) is a rhombus. Also, check side lengths (optional, but to confirm).

Step6: Calculate Side Lengths (Optional)

Length of \(WX\): \(W(7, -1)\), \(X(3, 6)\).
\(d_{WX} = \sqrt{(3 - 7)^2 + (6 - (-1))^2} = \sqrt{(-4)^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65}\).
Length of \(XY\): \(X(3, 6)\), \(Y(-1, -1)\).
\(d_{XY} = \sqrt{(-1 - 3)^2 + (-1 - 6)^2} = \sqrt{(-4)^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65}\).
Length of \(YZ\): \(Y(-1, -1)\), \(Z(3, -8)\).
\(d_{YZ} = \sqrt{(3 - (-1))^2 + (-8 - (-1))^2} = \sqrt{4^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65}\).
Length of \(ZW\): \(Z(3, -8)\), \(W(7, -1)\).
\(d_{ZW} = \sqrt{(7 - 3)^2 + (-1 - (-8))^2} = \sqrt{4^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65}\).
All sides are congruent, which also confirms it’s a rhombus (and since diagonals are perpendicular, it’s a rhombus; if diagonals were congruent, it would be a rectangle, but here diagonals are 8 and 14, not congruent, so not a rectangle, but a rhombus).

Answer:

The parallelogram \(WXYZ\) is a rhombus because its diagonals (horizontal \(WY\) and vertical \(XZ\)) are perpendicular, and all side lengths are equal (\(\sqrt{65}\) for each side). By the theorem that a parallelogram with perpendicular diagonals is a rhombus, \(WXYZ\) is a rhombus.