Dilate figures from the origin

A dilation centered at the origin maps each point $(x,y)$ to $(kx,ky)$ , where the scale factor $k$ may be an integer, fraction, or decimal; thus each coordinate and every distance from the origin is multiplied by $k$ . The image preserves angle measures and figure shape while changing size, supporting understanding of similarity and proportional relationships; this scope is limited to coordinate dilations centered at the origin, not arbitrary centers or more advanced transformations.

Detailed Explanation: 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.

Learn by doing: Dilate figures from the origin

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Dilations - Grid Image & Scale Factor (Origin) to Coordinate

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