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). Find the scale factor and explain each of your steps.

Answer 1

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:

`\sqrt[]{(c-a)^2+(d-b)^2}`

To find the length AB of the preimage, substitute the coordinates (a,b)=(9,4) and (c,d)=5,-4) into the formula:

`AB=\sqrt[]{(5-9)^2+(-4-4)^2}=\sqrt[]{(-4)^2+(-8)^2}=\sqrt[]{16+64}=\sqrt[]{80}`

To find the length A'B' of the image, substitute the coordinates (a,b)=(6,3) and (c,d)=(3,-3) into the formula:

`A^(\prime)B^(\prime)=\sqrt[]{(3-6)^2+(-3-3)^2}=\sqrt[]{(-3)^2+(-6)^2}=\sqrt[]{9+36}=\sqrt[]{45}`

Divide the length of the image by the length of the preimage to calculate the scale factor:

`(A^(\prime)B^(\prime))/(AB)=\frac{\sqrt[]{45}}{\sqrt[]{80}}=\sqrt[]{(45)/(80)}=\sqrt[]{(9)/(16)}=\frac{\sqrt[]{9}}{\sqrt[]{16}}=(3)/(4)`

Hence, the scale factor is 3/4.

The answer is 3/4.