Skill: Dilate figures from the origin

A dilation centered at the origin changes each point $(x,y)$ to

$$(x,y)\longrightarrow (kx,ky),$$

where $k$ is the scale factor. Multiply both coordinates by $k$.

• If $k>1$, the figure becomes larger.
• If $0<k<1$, the figure becomes smaller.
• Each image point stays on the same line from the origin as its original point.

Example

Dilate triangle $ABC$ by a scale factor of $2$, where

$$A(-3,1),\qquad B(1,2),\qquad C(0,-2).$$

Step 1: Write the rule.

Since $k=2$,

$$(x,y)\longrightarrow (2x,2y).$$

Step 2: Dilate each vertex.

For point $A(-3,1)$:

$$A'=(2(-3),2(1))=(-6,2).$$

For point $B(1,2)$:

$$B'=(2(1),2(2))=(2,4).$$

For point $C(0,-2)$:

$$C'=(2(0),2(-2))=(0,-4).$$

Step 3: State the image.

The dilated triangle has vertices

$$\boxed{A'(-6,2),\quad B'(2,4),\quad C'(0,-4)}.$$

To graph the image, plot these new points and connect them in the same order. The triangle has the same shape, but every distance from the origin is twice as large.