What is the scale factor of a dilation centered at the origin that maps the point (12, -8) to the point (-7.2, 4.8)?

Answer 1

The scale factor is a number by which you multiply the coordinates of the first point to get the coordinates of the second point. So if (x1, y1) is the first point and (x2, y2) is the second point in the dilation, and SF is the scale factor, then x1 * SF = x2 and y1 * SF = y2. You can find the scale factor by dividing the coordinates of the second point by the corresponding coordinates of the first point: SF = x2 ⁄ x1 and/or SF = y2 ⁄ y1. You should get the same value for the scale factor from both sets of coordinates. If you get something different, check your math again.

For the given problem, x1 = 12, y1 = -8, x2 = -7.2, and y2 = 4.8

SF = -7.2 ⁄ 12 = - 0.6 Check: (- 8) * (- 0.6) = 4.8

Answer 2

Grace,

Remember that the scale factor for a dilation can be understood as a factor that you multiply both the X-component and the Y-component of a point to dilate from the original point to the new point. For example if you had the point (1,2), and you scaled with a dilation factor of 3, you would multiply both the X component 1 and the Y component 2 to have a new dilated point represented as (3,6).

The same holds true for your problem. It may not seem clear given the odd X values and Y values of the new point, but there is an equivalent scale factor applied to both: -7.2 divided by 12 is -0.6, -4.8 divided by 8 is also -0.6. This means that since both components of the original point are multiplied by this same factor of -0.6, that must be our scale factor.

In other words, you multiply the values 12 and -8 by 0.6, and then as you have a negative scale factor you negate both values. 12 * -0.6 = -7.2, and -8 * -0.6 = 4.8. Negating the values can be understood as moving from the original quadrant (in this case quadrant IV as we have a positive X and negative Y value) to the opposite quadrant, quadrant II (negative X and positive Y value).

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Keith