Question

suppose you have ab on a coordinate plane located at a(-3,-4) and b(5,-4). under a dilation centered at (9,0), ab becomes a'b' with coordinates A'(6,-1) and B'(8,-1). What is the scale factor for this dilation

Answer 1

The distace between the center of dilation and point a is diven by:

$d=\sqrt{(9-(-3))^2+(0-(-4))^2} \\ \\ =\sqrt{(9+3)^2+(0+4)^2}=\sqrt{12^2+4^2} \\ \\ =\sqrt{144+16}=\sqrt{160}=\sqrt{16\times10}=4\sqrt{10}$

The distance between the center of dilation and point a' is given by:

$d=\sqrt{(9-6)^2+(0-(-1))^2} \\ \\ =\sqrt{3^2+(0+1)^2}=\sqrt{3^2+1^2} \\ \\ =\sqrt{9+1}=\sqrt{10}$

It can be seen that the distance from the center of dilation to the image is 1/4 times the distance from the centre of dilation to the preimage.

Therefore, the scale factor is 1/4.