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A triangle with vertices at [tex]$A (20, -30)$[/tex], [tex]$B (10, -15)$[/tex], and [tex][tex]$C (5, -20)$[/tex][/tex] has been dilated with a center of dilation at the origin. The image of [tex]B[/tex], point [tex]B'[/tex], has the coordinates [tex]$(2, -3)$[/tex]. What is the scale factor of the dilation?

A. [tex]\frac{1}{10}[/tex]
B. [tex]\frac{1}{5}[/tex]
C. 5
D. 10



Answer :

GinnyAnswer
To determine the scale factor of the dilation, we need to analyze the change in coordinates of point [tex]\( B \)[/tex] after dilation.

The original coordinates of point [tex]\( B \)[/tex] are:
[tex]\[ B (10, -15) \][/tex]

The coordinates of the image [tex]\( B' \)[/tex] after dilation are:
[tex]\[ B' (2, -3) \][/tex]

To find the scale factor, we compare the coordinates of [tex]\( B \)[/tex] and [tex]\( B' \)[/tex].

For the x-coordinates:
[tex]\[ B_x = 10 \][/tex]
[tex]\[ B'_x = 2 \][/tex]

The scale factor [tex]\( k \)[/tex] for the x-coordinate is calculated as:
[tex]\[ k_x = \frac{B'_x}{B_x} = \frac{2}{10} = 0.2 \][/tex]

For the y-coordinates:
[tex]\[ B_y = -15 \][/tex]
[tex]\[ B'_y = -3 \][/tex]

The scale factor [tex]\( k \)[/tex] for the y-coordinate is calculated as:
[tex]\[ k_y = \frac{B'_y}{B_y} = \frac{-3}{-15} = 0.2 \][/tex]

Since the scale factor must be consistent for both the x and y coordinates (i.e., the dilation is uniform in both directions), we have:
[tex]\[ k_x = k_y = 0.2 \][/tex]

Therefore, the scale factor of the dilation is:
[tex]\[ k = 0.2 \][/tex]

To convert 0.2 into a fraction for comparison with the given options:
[tex]\[ 0.2 = \frac{1}{5} \][/tex]

Thus, the scale factor of the dilation is:
[tex]\[ \boxed{\frac{1}{5}} \][/tex]

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