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 $k$ 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 with Scale Factor k

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 with Scale Factor k

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