Question

Line segment AB has endpoints A(2, 9) and B(5, 8) . A dilation, centered at the origin, is applied to AB¯¯¯¯¯ . The image has endpoints A′(43, 6) and B′(103, 163) . What is the scale factor of this dilation?
32
23
2
3

Answer 1

Answer:  The correct option is (B) $\dfrac{2}{3}.$

Step-by-step explanation:  Given that the co-ordinates of the end-points of a line segment AB are A(2, 9) and B(5, 8). After being dilated about the origin (0, 0), the co-ordinates of the end-points of image A'B' are $A^\prime\left(\dfrac{4}{3},6\right)$ and $B^\prime\left(\dfrac{10}{3},\dfrac{16}{3}\right).$

We are to find the scale factor of the dilation.

The scale factor of the dilation will be

$S=\dfrac{\textup{length of the image line}}{\textup{length of the original line}}.$

The lengths of the lines AB and A'B' are calculated using distance formula as follows:

$AB=\sqrt{(5-2)^2+(8-9)^2}=\sqrt{9+1}=\sqrt{10},\\\\\\A'B'=\sqrt{\left(\dfrac{10}{3}-\dfrac{4}{3}\right)^2+\left(\dfrac{16}{3}-6\right)^2\right)}=\sqrt{4+\dfrac{4}{9}}=\sqrt{\dfrac{40}{9}}=\dfrac{2}{3}\sqrt{10}~\textup{units}.$

Therefore, the required scale factor of dilation is

$S=\dfrac{A'B'}{AB}=\dfrac{\frac{2}{3}\sqrt{10}}{\sqrt{10}}=\dfrac{2}{3}.$

Thus, the scale factor of the dilation is $\dfrac{2}{3}.$

Option (B) is CORRECT.

Answer 2

The answer is 2/3 
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