Dilation

Resizing a Figure from a Center

Dilation is a geometric transformation that changes the size of an object (enlarging or shrinking) without changing its shape. Each point on the object is mapped to a new position based on a center of dilation and a scale factor.

Formal Definition of Dilation

Let $C$ be the center of dilation and let $k \neq 0$ be the scale factor. The dilation $D_{(C,k)}$ maps point $A$ to its image $A' = D_{(C,k)}(A)$ and satisfies $\vec{CA'} = k \cdot \vec{CA}$.

This means the vector from the center of dilation to the image is $k\text{ times}$ the vector from the center of dilation to the original point.

• If $|k| > 1$, the dilation enlarges the figure.
• If $0 < |k| < 1$, the dilation reduces the figure.
• If $k > 0$, the original point and its image are on the same side of the center of dilation.
• If $k < 0$, the original point and its image are on opposite sides of the center of dilation. The direction from the center to the image is opposite to the direction from the center to the original point.

Dilation with Respect to the Origin

If the center of dilation is the origin $O(0,0)$ and the scale factor is $k$, then for a point $A(x,y)$, its image $A'(x',y')$ is given by:

Dilating a Point with Respect to the Origin

If point $A(1,2)$ is dilated with respect to the origin $(0,0)$ by a factor of $2$, determine the image of the point.

Here, $x=1$, $y=2$, and $k=2$.

The center of dilation is $O(0,0)$.

The image is $A'(2,4)$.

Dilation with Respect to an Arbitrary Point

If the center of dilation is an arbitrary point $C(a,b)$ and the scale factor is $k$, then for a point $A(x,y)$, its image $A'(x',y')$ is given by:

The formula can be applied by shifting the coordinates so that $C$ becomes the origin, applying the dilation by factor $k$, and then shifting the coordinates back.

Dilating a Point with Respect to an Arbitrary Point

If point $C(5,2)$ is dilated with respect to point $P(2,3)$ by a factor of $2$, determine the image of the point.

Here, the point to be dilated is $C(5,2)$ so $x=5, y=2$.

The center of dilation is $P(2,3)$, so $a=2, b=3$.

The scale factor is $k=2$.

The image is $C'(8,1)$.

Exercises

The first two problems dilate one point, first about the origin and then about a different center. The third dilates a whole triangle with a negative factor, so the image lands on the opposite side of the center.

1. Determine the image of $B(2,5)$ under dilation $D_{(O,3)}$ (center at $O(0,0)$, factor $3$).
2. Determine the image of $B(2,5)$ under dilation with center $P(1,3)$ and factor $3$.
3. A triangle with vertices $A(1,1)$, $B(3,1)$, and $C(1,4)$ is dilated with respect to the origin $O(0,0)$ by a scale factor $k=-2$. Determine the coordinates of the image triangle $A'B'C'$!

Solutions

1. Point $B(2,5)$, center $O(0,0)$, $k=3$.

   $$x' = 3 \cdot 2 = 6$$

   $$y' = 3 \cdot 5 = 15$$

   Thus, the image is $B'(6,15)$.

2. Point $B(2,5)$, center $P(1,3)$, $k=3$. ($x=2, y=5, a=1, b=3$)

   $$\begin{aligned} x' &= 1 + 3(2 - 1) \\ &= 1 + 3(1) \\ &= 1 + 3 \\ &= 4 \end{aligned}$$

   $$\begin{aligned} y' &= 3 + 3(5 - 3) \\ &= 3 + 3(2) \\ &= 3 + 6 \\ &= 9 \end{aligned}$$

   Thus, the image is $B'(4,9)$.

3. Center $O(0,0)$, $k=-2$.

   • For $A(1,1)$: $A'(-2 \cdot 1, -2 \cdot 1) = A'(-2,-2)$.
   • For $B(3,1)$: $B'(-2 \cdot 3, -2 \cdot 1) = B'(-6,-2)$.
   • For $C(1,4)$: $C'(-2 \cdot 1, -2 \cdot 4) = C'(-2,-8)$.

   The coordinates of the image triangle are: $A'(-2,-2)$, $B'(-6,-2)$, $C'(-2,-8)$.