Question

What is the scale factor of a pair of figures whose preimage has coordinates(2,3), (3, 1), and (4,5), and whose image has coordinates (10, 15). (15,5), and (20, 25).

Answer 1

Let:

$\begin{gathered} (x1,y1)=(2,3) \\ (x2,y2)=(3,1) \\ d1=\sqrt[]{(x2-x1)^2+(y2-y1)^2} \\ d1=\sqrt[]{(3-2)^2+(1-3)^2} \\ d1=\sqrt[]{1+4}=\sqrt[]{5} \end{gathered}$$\begin{gathered} (x1^{\prime},y1^{\prime})=(10,15) \\ (x2^{\prime}y2^{\prime})=(15,5) \\ d2=\sqrt[]{(x2^{\prime}-x1^{\prime})^2+(y2^{\prime}-y1^{\prime})^2} \\ d2=\sqrt[]{(15-10)^2+(5-15)^2} \\ d2=\sqrt[]{25+100}=5\sqrt[]{5} \end{gathered}$

k = scale factor

so:

$\begin{gathered} k\cdot d1=d2 \\ k\sqrt[]{5}=5\sqrt[]{5} \end{gathered}$

solve for k:

$\begin{gathered} k=\frac{5\sqrt[]{5}}{\sqrt[]{5}} \\ k=5 \end{gathered}$

The scale factor is 5