Chapter 5.6, Problem 36PPE

To determine

ABCD is a rectangle.

Explanation of Solution

Given data:

  $A(-3,3),B(-1,-2),C(4,0),D(2,5)$

Formula used:

Slope intercept form:

  $\begin{array}{l}y=mx+C\\ m=\frac{y_2-y_1}{x_2-x_1}\end{array}$

Calculation:

For line AB,

  $\begin{array}{l}y=mx+C\\ m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-2-3}{-1-(-3)}\\ m=-\frac{5}{2}\\ y=-\frac{5}{2}x+C\\ -2=-\frac{5}{2}(-1)+C\\ C=-\frac{9}{2}\end{array}$

Similarly for,

Line BC

  $y=\frac{2}{5}x-\frac{8}{5}$

Line CD

  $y=-\frac{5}{2}x-10$

Line DA

  $y=\frac{2}{5}x-\frac{21}{5}$

As the line AB and CD have same slope so they are parallel to each other and similarly BC and AD are parallel to each other.

As the lines AB and BC have opposite slope so they are perpendicular to each other similarly BC ⊥ CD and CD ⊥ DA and DA ⊥ AB.

So ABCD is a rectangle.

Conclusion:

ABCD is a rectangle.