Use the vertices of the rectangle to determine the slopes of the lines and choose one vertex for each line to write the equation.
B
b
Compare the slopes of the lines. What did you notice?
C
c
What did you notice in Part B?
A
a
Line AB: y-1=1/3(x+3)
Line BC: y-3=-3(x-3) Line CD: y=1/3(x-4) Line DA: y+2=-3(x+2)
B
b
They have the same slopes.
C
c
y+4=-7x
Practice makes perfect
a
Let's identify a point on each line and the slope for each side of the rectangle ABCD. Then we can write a point-slope form equation for each of them.
Let's start with the line that passes through points A and B. We know that the coordinates of points A and B are (-3,1) and (3,3), respectively. We can use these points to find the slope of the line using the Slope Formula.
The slope of the line AB is 13. Now, we can use the point A( -3, 1) to write the equation in point-slope form. In point-slope form, (x_1,y_1) is a point on the line and m is the slope.
y-y_1=m(x-x_1)
⇒
y- 1= 13(x-( -3))
We have three more lines. If we follow the same steps, we will have the following table and equations.
Line
AB
BC
CD
DA
Points
A( -3,1),B( 3,3)
B( 3,3),C( 4,0)
C( 4,0),D( -2,-2)
D( -2,-2),A( -3,1)
Slope Formula
m=3- 1/3-( -3)
m=0- 3/4- 3
m=-2- 0/-2- 4
m=1-( -2)/-3-( -2)
Calculation
m=1/3
m=-3
m=1/3
m=-3
Reference Point
A(-3,1)
B(3,3)
C(4,0)
D(-2,-2)
Equation
y-1=1/3(x+3)
y-3=-3(x-3)
y=1/3(x-4)
y+2=-3(x+2)
b
Did you notice in the table that the lines AB and CD have the same slope? Lines BC and DA also have the same slope. By looking at the rectangle, we can see that these pairs of lines appear to be parallel. We can say that parallel lines have the same slope.
c
Based on our conjecture in Part B, if a line is parallel to y-9= -7(x+3), it should also have a slope of -7. Since we have the slope and a point that the line passes through, ( 0, -4), we can write the equation of the line in point-slope form.
