Verify that parallelogram WXYZ with vertices W(-10, -2), X(-18,12), Y(-2,10), and Z (6,-4) is a rhombus by showing that it is a parallelogram with perpendicular diagonals.

The formula of the slope of a line is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
[tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] are two points on the line

∵ WXYZ is a parallelogram

∵ W (-10 , -2) , X (-18 , -12) , Y (-2 , 10) , Z (6 , -4)

∵ Its diagonals are WY and XZ

- Find the slope of WY

∵ W = (-10 , -2) and Y = (-2 , 10)

∴ [tex]x_{1}[/tex] = -10 and [tex]x_{2}[/tex] = -2

∴ [tex]y_{1}[/tex] = -2 and [tex]y_{2}[/tex] = 10

∴ [tex]m_{WY}=\frac{10--2}{-2--10}=\frac{12}{8}[/tex]

- Divide up and down by 4 to reduce it to its simplest form

∴ [tex]m_{WY}=\frac{3}{2}[/tex]

- Find the slope of XZ

∵ X = (-18 , 12) and Z = (6 , -4)

∴ [tex]x_{1}[/tex] = -18 and [tex]x_{2}[/tex] = 6

∴ [tex]y_{1}[/tex] = 12 and [tex]y_{2}[/tex] = -4

∴ [tex]m_{XZ}=\frac{-4-12}{6--18}=\frac{-16}{24}[/tex]

- Divide up and down by 8 to reduce it to its simplest form

∴ [tex]m_{XZ}=\frac{-2}{3}[/tex]

∵ [tex](m_{WY}).(m_{XZ})=(\frac{3}{2})(\frac{-2}{3})=\frac{-6}{6}=-1[/tex]

∴ WY ⊥ XZ

∴ The diagonals of the parallelogram are perpendicular

∴ The parallelogram WXYZ is a rhombus

The parallelogram WXYZ is a rhombus because its diagonals are perpendicular

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You can learn more about parallelograms in brainly.com/question/4459688

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