Exercise 36 Page 335

To begin we will plot the given points. Then we will draw the lines that construct ABCD.

Based on the graph, we can see that for ABCD to be a rectangle each of the following must be true.

1. Lines AB and DC must be parallel.
2. Lines AD and BC must be parallel.
3. Lines AB and BC must be perpendicular.
4. Lines AD and DC must be perpendicular.

   For two lines to be parallel they must have equal slopes. Also, for two lines to be perpendicular their slopes must multiply to equal - 1.

   Finding Slopes

   We will begin by finding the slopes of each line using the given coordinates and the Slope Formula.

   Line	Points	y_2-y_1/x_2-x_1	Slope
   AB	(- 3,3), (- 1,- 2)	- 2-3/- 1-(- 3)	\text{-} \dfrac {5}{2}
   BC	(- 1,- 2), (4,0)	0-(- 2)/4-(- 1)	\dfrac {2}{5}
   DC	(2,5), (4,0)	0-5/4-2	\text{-} \dfrac {5}{2}
   AD	(- 3,3), (2,5)	5-3/2-(- 3)	\dfrac {2}{5}

   We can now use these slopes to determine which lines are parallel or perpendicular.

   Parallel Lines

   From the table we can see the lines that have the same slope.

   1. Line AB and Line DC have the same slope. Thus, they are parallel.
   2. Line BC and Line AD have the same slope. Thus, they are parallel.

   Perpendicular Lines

   We will now seek to prove that the adjacent sides of the rectangle are perpendicular. We will test the slopes of lines AB and BC.
   m_{AB} \cdot m_{BC}\stackrel{?}=\text{-}1

   SubstituteII

   m_{AB}={\color{#0000FF}{ \text{-} \dfrac {5}{2}}}, m_{BC}={\color{#009600}{\dfrac {2}{5} }}

   {\color{#0000FF}{\text{-} \dfrac {5}{2}}} \cdot {\color{#009600}{\dfrac {2}{5}}} \stackrel{?}=\text{-}1

   MultFrac

   Multiply fractions

   \text{-} \dfrac{10}{10}\stackrel{?}=\text{-}1

   CalcQuot

   Calculate quotient

   - 1=-1 ✓
   AB and BC are perpendicular. Now we will test the slopes of lines AD and DC.
   m_{AD}\cdot m_{DC}\stackrel{?}=\text{-}1

   SubstituteII

   m_{AD}={\color{#0000FF}{\dfrac {2}{5}}}, m_{DC}={\color{#009600}{\text{-} \dfrac {5}{2}}}

   {\color{#0000FF}{\dfrac {2}{5}}} \cdot {\color{#009600}{\text{-} \dfrac {5}{2}}} \stackrel{?}=\text{-}1

   MultFrac

   Multiply fractions

   \text{-} \dfrac{10}{10}\stackrel{?}=\text{-}1

   CalcQuot

   Calculate quotient

   - 1=-1 ✓
   AD and DC are perpendicular as well. As it has been proven that the opposite sides are parallel and the adjacent sides are perpendicular, we can conclude that ABCD is a rectangle.