Dilate the point by multipling their coordinates by 12.
B
b
Use the Slope Formula to get the slope of the lines. Do you notice any difference?
A
a
2(A'B')=AB
B
b
AB and A'B' have the same slope but AB has a y-intercept that is 0.5 units above the y-intercept of A'B'.
Practice makes perfect
a
To dilate the line through the points A(0,1) and B(1,2) we will start by graphing the points and the line through them in a coordinate system.
When dilating points we should multiply the coordinates with the scale factor. In this case, the scale factor is k= 12 and multiplying something by 12 is the same as dividing it by 2.
(x,y) &→ ( x2, y2)
A(0,1) &→ A'(0,0.5)
B(1,2) &→ B'(0.5,1)
We will now mark the images and a line through them in the system.
We should now investigate the lengths and to do that we must calculate them. This can be done using the Distance Formula.
Segment
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Distance
AB
d = sqrt(( 1- 0)^2 + ( 2- 1)^2)
sqrt(2)
A'B'
d = sqrt(( 0.5- 0)^2 + ( 1- 0.5)^2)
1/2sqrt(2)
Therefore, the length of A'B' is half as long as AB.
2(A'B')=AB
b
From the graph, we can see that the line AB is above A'B'.
To compare the lines we should calculate their slope. We will start with the preimage, AB using the Slope Formula.
Line
m = y_2-y_1/x_2-x_1
Slope
AB
m = 2- 1/1- 0
1
A'B'
m = 1- 0.5/0.5- 0
1
The lines have the same slope, m=1. Even though they have the same slope they do not intersect the axis on the same coordinates. Let's take a closer look at where they intercept the y-axis.
The preimage intercepts at (0,1) and the image and (0,0.5). This gives us a vertical difference between the y-intercepts of 1-0.5=0.5 units.
