Question

The endpoints of AB are A(9, 4) and B(5, –4). The endpoints of its image after a dilation are A'(6, 3) and B'(3, –3). Explain how to find the scale factor.

Answer 1

AB can be represented by <-4,-8> which just means to walk from point A to point B we go -4 units in the x direction and -8 units in the y direction. Similarly, A'B' can be represented as <-3,-6>. Ok, the scale factor is what we have to multiply AB by to get A'B':
s<-4,-8>=<-3,-6>
so,
-4s=-3
s=3/4

Answer 2

Another way to do this is to find the length of your line AB.
\[AB=\sqrt{\Delta x^2+\Delta y^2}=\sqrt{(5-9)^2+(-4-4)^2}=4\sqrt{5}\]Now we find the length of A'B' and compare them:\[A'B'=\sqrt{(3-6)^2+(-3-3)^2}=3\sqrt{5}\]ok...the scale factor just changes the length from 4sqrt(5) to 3sqrt(5). Calling s the scale factor:\[(4\sqrt{5})s=3\sqrt{5}\]Thus\[s=\frac{3}{4}\]as before