Dilations and Scale Factors

A dilation is a type of geometric transformation that changes the size of a figure but keeps its shape exactly the same. Imagine zooming in or out on a picture on your phone; that is a real-world example of a dilation.

Understanding the Scale Factor ($k$)

Every dilation requires a center point (usually the origin, $(0,0)$) and a scale factor, commonly represented by the variable $k$. The scale factor tells you how much the figure will stretch or shrink:

• Enlargement ($k > 1$): If the scale factor is greater than $1$, the figure gets larger.
• Reduction ($0 < k < 1$): If the scale factor is a fraction or decimal between $0$ and $1$, the figure gets smaller.
• Congruence ($k = 1$): If the scale factor is exactly $1$, the figure stays the exact same size.

Dilating Points from the Origin

When the center of dilation is the origin $(0,0)$, calculating the new coordinates of a figure is very straightforward. You simply multiply both the $x$- and $y$-coordinates of the original point by the scale factor $k$.

The mathematical rule is: $(x, y) \rightarrow (kx, ky)$

What Changes and What Stays the Same?

Because a dilation creates a similar figure (same shape, different size), it has specific properties:

• Angles: Angle measures stay exactly the same.
• Side Lengths: Every side length is multiplied by the scale factor $k$.
• Area: The area of the figure does not just multiply by $k$. Instead, the area is multiplied by $k^2$ (the square of the scale factor).

Example Problems

Example 1: Dilate a point Dilate the point $(3, 6)$ from the origin with a scale factor of $\frac{1}{3}$.

• Solution: Multiply both coordinates by $\frac{1}{3}$. $x = 3 \times \frac{1}{3} = 1$ $y = 6 \times \frac{1}{3} = 2$ The new point is $(1, 2)$.

Example 2: Enlarge a shape Enlarge a triangle with a vertex at $(4, -2)$ with a scale factor of $2$ from the origin.

• Solution: Multiply the coordinates by $2$. $x = 4 \times 2 = 8$ $y = -2 \times 2 = -4$ The new vertex is $(8, -4)$. You would repeat this for all vertices of the triangle.

Example 3: Area changes How does the area of a figure change after a dilation with a scale factor of $3$?

• Solution: Since side lengths are multiplied by $3$, the area is multiplied by $3^2$. $3^2 = 9$ The new area is $9$ times larger than the original area.