It is given that the line segment AB is dilated to give another line segment A'B'.
Since it is a dilation, the length of the image will be a multiple of the length of the preimage.
To find the scale factor, divide the length of the image by the length of the preimage.
Recall that the length of a line segment with endpoints (a,b) and (c,d) is given as:
[tex]\sqrt[]{(c-a)^2+(d-b)^2}[/tex]
To find the length AB of the preimage, substitute the coordinates (a,b)=(9,4) and (c,d)=5,-4) into the formula:
[tex]AB=\sqrt[]{(5-9)^2+(-4-4)^2}=\sqrt[]{(-4)^2+(-8)^2}=\sqrt[]{16+64}=\sqrt[]{80}[/tex]
To find the length A'B' of the image, substitute the coordinates (a,b)=(6,3) and (c,d)=(3,-3) into the formula:
[tex]A^{\prime}B^{\prime}=\sqrt[]{(3-6)^2+(-3-3)^2}=\sqrt[]{(-3)^2+(-6)^2}=\sqrt[]{9+36}=\sqrt[]{45}[/tex]
Divide the length of the image by the length of the preimage to calculate the scale factor:
[tex]\frac{A^{\prime}B^{\prime}}{AB}=\frac{\sqrt[]{45}}{\sqrt[]{80}}=\sqrt[]{\frac{45}{80}}=\sqrt[]{\frac{9}{16}}=\frac{\sqrt[]{9}}{\sqrt[]{16}}=\frac{3}{4}[/tex]
Hence, the scale factor is 3/4.
The answer is 3/4.
