Dilation

Understanding Dilation

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

Given a point $C$ as the center of dilation and a scale factor $k \neq 0$. The dilation of a point $A$ with respect to center $C$ by a factor $k$, denoted as $D_{(C,k)}$, is a transformation that maps $A$ to $A' = D_{(C,k)}(A)$ such that $\vec{CA'} = k \cdot \vec{CA}$.

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 $|k| > 1$, it is an enlargement.
• If $0 < |k| < 1$, it is a reduction.
• 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 (and the image is inverted).

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 $O(0,0)$.

Thus, 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:

This can be interpreted as: translate the system so that $C$ becomes the origin, perform the dilation by factor $k$, and then translate 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 $k=2$.

Thus, the image is $C'(8,1)$.

Exercises

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 $A'B'C'$!

Key Answers

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

   $$x' = 1 + 3(2 - 1) = 1 + 3(1) = 1 + 3 = 4$$

   $$y' = 3 + 3(5 - 3) = 3 + 3(2) = 3 + 6 = 9$$

   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: $A'(-2,-2)$, $B'(-6,-2)$, $C'(-2,-8)$.