Chapter 6.4, Problem 9CYU

To determine

To find: The type of quadrilateral which is formed by the given coordinates.

Answer to Problem 9CYU

The given points are the vertices of a rectangle.

Explanation of Solution

Given information: $A\left(4,3\right),B\left(4,-2\right),C\left(-4,-2\right),D\left(-4,3\right)$

Formula used: $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Calculation: The given vertices are plotted below.

Applying the distance formula to calculate the length of the sides,

  $\begin{array}{c}AB=\sqrt{{\left(4-4\right)}^2+{\left(3-\left(-2\right)\right)}^2}\\ =\sqrt{5^2}\\ =5\end{array}$

  $\begin{array}{c}BC=\sqrt{{\left(4-\left(-4\right)\right)}^2+{\left(-2-\left(-2\right)\right)}^2}\\ =\sqrt{8^2}\\ =8\end{array}$

  $\begin{array}{c}CD=\sqrt{{\left(-4-\left(-4\right)\right)}^2+{\left(-2-3\right)}^2}\\ =\sqrt{5^2}\\ =5\end{array}$

  $\begin{array}{c}DA=\sqrt{{\left(-4-4\right)}^2+{\left(3-3\right)}^2}\\ =\sqrt{8^2}\\ =8\end{array}$

Since, $AB=CD\&BC=DA$

Hence, the given points are the vertices of a rectangle.