A coordinate plane with four quadrants shows an x axis capped with arrows ranging from negative 3 to 12 in 1 unit increments and a y axis capped with arrows ranging from 15 to negative 4 in 1 unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted in quadrant 1. Point upper A is plotted as left parenthesis 9 comma 9 right parenthesis and point upper B is plotted as left parenthesis 6 comma 6 right parenthesis. The points are plotted on a line segment. Point upper A prime is plotted as left parenthesis 3 comma 3 right parenthesis and upper B prime is plotted as left parenthesis 2 comma 2 right parenthesis. The points are plotted on a line segment.

Question

Determine the scale factor used in the dilation of line segment  AB, with a center point of dilation at the origin.

3

−3

−13

13

Answer 1

To determine the scale factor used in the dilation of line segment $AB$ (from points $A$ and $B$) to line segment $A^{\prime }{B}^{\prime }$ (from points $A^{\prime }$ and $B^{\prime }$), we can use the coordinates of the points.

1. Identify the coordinates of the points:

   • $A(9,9)$
   • $B(6,6)$
   • $A^{\prime }(3,3)$
   • $B^{\prime }(2,2)$

2. Find the distances (lengths) of line segments $AB$ and $A^{\prime }{B}^{\prime }$:

   • The distance $AB$ can be calculated using the distance formula: $$\text{Distance }AB=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$ For points $A(9,9)$ and $B(6,6)$: $$AB=\sqrt{(6-9{)}^2+(6-9{)}^2}=\sqrt{(-3{)}^2+(-3{)}^2}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$$

   • For points $A^{\prime }(3,3)$ and $B^{\prime }(2,2)$: $$A^{\prime }{B}^{\prime }=\sqrt{(2-3{)}^2+(2-3{)}^2}=\sqrt{(-1{)}^2+(-1{)}^2}=\sqrt{1+1}=\sqrt{2}$$

3. Find the scale factor (k): The scale factor $k$ of the dilation can be found by comparing the lengths of the two line segments: $$k=\frac{\text{Length of }{A}^{\prime }{B}^{\prime }}{\text{Length of }AB}=\frac{\sqrt{2}}{3\sqrt{2}}=\frac{1}{3}$$

4. Because the dilation occurs about the origin, the scale factor needs to be taken into account: The ratio $\frac{A^{\prime }{B}^{\prime }}{AB}=\frac{1}{3}$ means that the scale factor is $\frac{1}{3}$, not one of the provided multiple-choice answers (3, -3, -13, 13).

Thus, if the question indeed asks for the dilation factor to a dilation that shortens the segment by a factor of $\frac{1}{3}$, the correct answer in the context of the original answers provided is not available.

If we interpret the negative values as potentially indicating opposite orientation or reflection based on the context of dilation about the origin, it would not fit as a standard scale factor. The scale factor generally would just be a positive fraction, going from larger to smaller as described.

So given the provided answers don't match $\frac{1}{3}$, if one must choose, the answer is not fitting into the options; it should be retained as is: $\frac{1}{3}$.

Therefore, none of the provided options (3, -3, -13, 13) is correct in this interpretation, since $\frac{1}{3}$ is what we derived. But if in a positive sense, the dilation is characterized as such, $3$ could potentially mislead senses of magnitude increase if considering magnitudes only or similarly $13$ if enacting arbitrary increases with no contextual geometric relevance.

The scale factor based on practical calculation leading to uniform contraction of segment is watermarked at $\frac{1}{3}$ if looking for strict relevance, feeling none here matches as described above.