Dilation in Geometry

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Dilation is a process used to change the size of a shape to become either larger or smaller. Unlike other transformations, dilation often does not cause the shape to flip, rotate, or change position on the $xy$-plane. The only time dilation does cause a rotation is if it is multiplied by a negative scale factor. When this happens, the shape becomes larger or smaller and rotates $180°$ about the origin. Dilation also does not cause the shape to become more narrow or thicker; it keeps the proportions of shapes the same and changes only their size. This is because all dilations have an associated scale factor, called $k$, which describes to what degree the size of the shape will change. For example, if $k=\frac{1}{2}$, the shape will shrink to half its original size, but if $k=4$, the shape will grow to $4$ times its original size.

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Dilating a triangle, as with any shape, will cause it to grow or shrink in size. It will not cause the triangle to rotate, flip, or distort. When $k>1$, the triangle will grow as each of the $3$ points of the triangle move away from the origin. On the other hand, when $0<k<1$, the triangle will shrink as all $3$ points move toward the origin.

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Two things are needed to dilate: an original shape and a scale factor $k$. Write down the coordinates of each point of the original shape and label them. To find the points of the new, dilated shape, simply multiply each of the original coordinates by $k$, then connect the dots!

For example, the corners of the red polygon ABCD have the following coordinates:

$A:(-4,4)$

$B:(0,-2)$

$C:(4,0)$

$D:(8,4)$

To dilate polygon ABCD with scale factor $k=\frac{1}{2}$, we multiply each of the points’ $x$– and $y$-coordinates by $\frac{1}{2}$. These new points will be called A’, B’, C’, and D’.

$A^{\prime }:(-4\times \frac{1}{2},4\times \frac{1}{2})=(-2,2)$

$B^{\prime }:(0\times \frac{1}{2},-2\times \frac{1}{2})=(0,-1)$

$C^{\prime }:(4\times \frac{1}{2},0\times \frac{1}{2})=(2,0)$

$D^{\prime }:(8\times \frac{1}{2},4\times \frac{1}{2})=(4,2)$

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A dilated shape can be bigger or smaller, depending on the scale factor $k$. When $k>1$, the shape will grow as all of its points move away from the origin. On the other hand, when $0<k<1$, the shape will shrink as its points move toward the origin.

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Even if the scale factor $k$ is negative, the same rule applies: the coordinates of dilated points can be found by multiplying the original coordinates by $k$. For example, the red triangle below has corners at points $(-1,-2),(2,2),$ and $(3,0)$. The green triangle is its dilation with $K=-1$, so the dilated corners are at points $(1,2),(-2,-2),$ and $(-3,0)$, which can all be found by multiplying the coordinates of the original points by $-1$.

Sometimes it can be helpful to draw lines that connect the original points to the origin, and then follow those lines the appropriate distance past the origin to mark the dilated points.

Notice that when you dilate by a negative scale factor, the figure is rotated $180°$ about the origin.

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Because the coordinates of dilated points are all multiplied by the same $k$, the proportions of the shape stay the same. For example, if done correctly, a square will not dilate into a long and skinny rectangle, and a trapezoid will not dilate into a rhombus.

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Any time k is positive, orientation is not changed. The shape will still “face” the same direction. However, if $k$ is negative, the dilated shape flips over across the origin, as shown below. This “flip” is the shape rotating by $180°$ about the origin.

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No, dilation does not preserve distance. Because each point changes during dilation, and they all get closer to or further away from the origin, they also get closer to or further away from each other based on the scale factor.

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The scale factor of dilation is the degree to which a shape is grown or shrunk, and is denoted $k$. When $k>1$, the shape will grow as all of its points move away from the origin. On the other hand, when $0<k<1$, the shape will shrink as its points move toward the origin. If $k$ were to equal $1$, then the shape would not change at all. If $k$ were to equal $0$, then all points would go straight to $0$ and the shape would disappear. These relationships come from the fact that the coordinates of dilated points are the product of the original coordinates multiplied by $k$.

$A=(x,y)$ implies $A^{\prime }=(K\times x,K\times y)$

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The center of dilation is a fixed point on the plane that is the point of reference for where the shape may shrink towards or grow away from. It is most common for the center of dilation to be the origin, but it may be at other places on the plane. For example, the illustration below shows a dilation of a triangle from a center at $(-4,-9)$.