Exercise 8 Page 297

Let's label the vertices with their coordinates.

If this quadrilateral is a rectangle, three conditions have to be meet:

1. Opposite sides have to be parallel: AD∥ BC and AB∥ DC
2. Opposite sides have to have equal lengths: AD≅BC and AB∥ DC
3. Adjacent sides have to be perpendicular: AD⊥ AB and AD⊥ DC.

   If opposite sides are parallel, they have the same slope. We can investigate this by using the Slope Formula.

   Side	Points	y_2-y_1/x_2-x_1	m
   AB	( 4,7), ( 2,5)	7- 5/4- 2	1
   DC	( 7,4), ( 5,2)	7- 5/4- 2	1
   AD	( 5,2), ( 2,5)	2- 5/5- 2	- 1
   BC	( 7,4), ( 4,7)	4- 7/7- 4	- 1

   Since opposite sides have the same slope, they are parallel. Next, we have to prove that opposite sides have equal lengths. For this purpose, we use the Distance Formula.

   Distance	Points	sqrt((x_2-x_1)^2+(y_2-y_1)^2)	d
   AB	( 4,7), ( 2,5)	sqrt(( 4- 2)^2+( 7- 5)^2)	sqrt(8)
   DC	( 7,4), ( 5,2)	sqrt(( 7- 5)^2+( 4- 2)^2)	sqrt(8)
   AD	( 5,2), ( 2,5)	sqrt(( 5- 2)^2+( 2- 5)^2)	sqrt(18)
   BC	( 7,4), ( 4,7)	sqrt(( 7- 4)^2+( 4- 7)^2)	sqrt(18)

   As we can see, opposite sides also have the same length. Therefore, we know that AB≅ DC and AD≅ BC.

   

   Let's also investigate the last criteria. Perpendicular lines have slopes that are opposite reciprocals of each other. This means the product of their slopes equals - 1. Let's test this.

   Sides	Slopes	m_1* m_2=- 1	Simplify	Perpendicular?
   AB, AD	m_(AB)= 1, m_(AD)= - 1	1( - 1)? =- 1	- 1 = - 1	Yes
   DC, AD	m_(DC)= 1, m_(AD)= - 1	1( - 1)? =- 1	- 1 = - 1	Yes

   AD is perpendicular to both AB and DC. Additionally, since AD is parallel to BC which means AB and DC are perpendicular to BC as well.

   

   Now we have proven that ABCD is a rectangle.