Question

How does dilation affect a figure on a coordinate plane?

Answer 1

Hint: Here we will illustrate a dilation in a coordinate plane, by demonstrating how to use the origin as the center of the dilation and the given scale factor to find the coordinates of the vertices of the image.

Complete step-by-step solution:
Dilation on a coordinate plane:
Objects like pictures or movies can be dilated objects in the coordinate plane can also be dilated. Dilation is an enlargement or reduction of an object by a scale factor and with a center of dilation. The scale factor refers to the change in size. The center of dilation is a point about which we are dilating an object. Usually, the center of dilation is the origin $ \left( {0,0} \right)$ .
Assuming the center of dilation is at point $ \left( {0,0} \right)$ on the coordinate plane and a factor of dilation $ f$ , of a point $ A\left\{ {x,y} \right\}$ will be transformed into point $ A'\left\{ {fx,fy} \right\}$ .
Dilation or scaling is the transformation of the plane according to the following rules:
There is a fixed point $ O$ on a plane or in space that is called the center of scaling.
There is a real number $ f \ne 0$ that is called the factor of scaling.
The transformation of any point $ P$ into its image $ P'$ is done by shifting its position along the line $ OP$ in such a way that the length of $ OP'$ equals to the length of $ OP$ multiplied by a factor$ \left| f \right|$ , that is$ \left| {OP'} \right| = \left| f \right|.\left| {OP} \right|$ . Since there are two candidates for point $ P'$ on both sides from center of scaling $ O$ , the position is chosen as follows: for $ f > 0$ both $ P$ and $ P'$ are supposed to be on the same side from center $ O$ otherwise, if $ f < 0$ , they are supposed to be on opposite sides of center $ O$ .
It can be proven that the image of a straight line $ l$ is a straight line $ l'$ . Segment $ AB$ is transformed into a segment $ A'B'$ , where $ A'$ is an image of point $ A$ and $ B$ is an image of point $ B$ .
Dilation preserves parallelism among lines and angles between them.
The length of any segment $ AB$ changes according to the same rule above$ \left| {A'B'} \right| = f.\left| {AB} \right|$ .
Using the coordinates the above properties can be expressed in the following form.
Assuming the center of dilation is at point $ \left\{ {0,0} \right\}$ on the coordinate plane and a factor of dilation $ f$ , a point $ A\left\{ {x,y} \right\}$ will be transformed into point $ A'\left\{ {fx,fy} \right\}$ . If the center of dilation is at point $ C\left\{ {p,q} \right\}$ , the point $ A\left\{ {x,y} \right\}$ will be transformed by the dilation into $ A'\left\{ {p + f(x - p),q + f(y - q)} \right\}$ .

Note: An object is enlarged if the scale factor is greater than $ 1$ . An object is reduced if the scale factor is a fraction, less than$ 1$ . To find the dilated image, we first must know the coordinates of our original image. Then we simply multiply the coordinate by the scale factor to find the dilated image.