Question

Write a coordinate proof of the following theorem:"If a quadrilateral is a kite, then its diagonals are perpendicular."(image attached)thank you ! :)

Answer 1

Explanations:

Given the following coordinate points from the kite

`\begin{gathered} W=(a,4b) \\ X=(2a,b) \\ Y=(a,0) \\ Z=(0,b) \end{gathered}`

For the diagonals to be perpendicular the product of the distance WY and XZ must be zero that is;

`\vec{WY}\cdot\vec{XZ}=0`

Determine the coordinate point WY

`\begin{gathered} \vec{WY}=[(a-a,4b-0)] \\ \vec{WY}=(0,4b) \end{gathered}`

Determine the coordinate point XZ

`\begin{gathered} \vec{XZ}=[(2a-0),(b-b)] \\ \vec{XZ}=(2a,0) \end{gathered}`

Take the dot product of the coordinates

`\begin{gathered} \vec{WY}\cdot\vec{XZ}=(0,4b)\cdot(2a,0) \\ \vec{WY}\cdot\vec{XZ}=[(0)(2a)+(4b)(0))] \\ \vec{WY}\cdot\vec{XZ}=(0,0)=\vec{0} \end{gathered}`

Since the dot product of the coordinates is a zero vector, hence its diagonals are perpendicular.