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XYZ has coordinates X(12,9), Y(-1,4), and Z(8,16). After a dialation with a center at 0, X' is (3,1.8). What is the scale factor? What coordinate is Y' & Z'?

I really dont get this, and need someone professional to explain this to me.

Hi, Courtney.

I can see why you are confused by this problem! But first, let's review dilations.

A dilation is a transformation that stretches or shrinks the original figure to produce a similar figure; the new figure is the same shape as the original, but a different size.

The firgure is enlarged or reduced with respect to a fixed point called the center of dilation.

Remember ratios of similar figures? The scale factor is the ratio of the length of a side of the new figure to the corresponding side length of the original figure. For example, if the scale factor is 1/2, each side of the new figure will be 1/2 the size of the original. Also, when the scale factor is between 0 and 1, the dilation is a reduction. A scale factor greater than one would result in an enlargement.

Further, the scale factor is also the ratio of the distances from the center of dilation to the corresponding points of the figure (new distance to original distance).

If the center of dilation is the origin, then the coordinates are multiplied by the scale factor: (x,y) -> (kx, ky) where k is the scale factor.

To solve a problem like the one you presented, determine the scale factor by dividing the coordinates of X' by the corresponding coordinates of X. Then, multiply the other coordinates by that scale factor.

What is confusing about the problem you presented is that the scale factor is not consistent: 3/12 = .25, 1.8/9 = .2. If you plot them on the coordinate plane, they do not line up with the origin.  I'm guessing that there is a typo in either your original problem or the way that you entered it here.

Hope the explanation helps!

Hi Courtney,

You can think of the points X, Y, Z as vectors originating from the origin.  Each of these "vectors" would have a magnitude associated with it, found by using the Pythagorean Theorem.

For example, the magnitude of X would be sqrt(12^2 + 9^2) = 15.

After the dilation, the magnitude of X became sqrt(3^2 + 1.8^2) = 3.50.

So the dilation caused X to shrink by a factor of 3.50/15 = 0.233.  This factor is the scale factor.

In addition to being scaled, the graph was also rotated.  You can tell because the angle that X makes with the x-axis [arctan(9/12) = 0.644] is not the same as the angle that X' makes with the x-axis [arctan(1.8/3) = 0.540].  The difference in these angles is the amount that the graph was rotated by [0.540 - 0.644 = -0.103 radians].

To find Y' and Z', simply take the Y and Z vectors through the same scaling and rotating transformations.  I will walk you through how to do this for Y below.

To scale Y, multiply its coordinates by the scale factor.

Yscaled = (0.233*-1, 0.233*4) = (-0.233, 0.932)

To rotate Y, left-multiply Yscaled by the rotation matrix for -0.103 radians.

Y'x = -0.233*cos(-0.103) - 0.932*sin(-0.103) = -0.136

Y'y = -0.233*sin(-0.103) + 0.932*cos(-0.103) = 0.951

This gives Y' = (-0.136, 0.951).

Hope this helps!

Another way to perform the rotation is to think of Yscaled as a complex number: -0.233 + 0.932i.  Then rotating Yscaled counterclockwise by ? would be equivalent to multiplying it by ei?.   The real part of the product would be Y'x, and the imaginary part would be Y'y.

11/29/2012

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