Question

1. write a coordinate rule for the dilation.

rule:

Explanation:

Step1: Find the coordinates of corresponding points

Let's take point \(B(0,4)\) and \(B'(0,2)\), point \(C(4,0)\) and \(C'(2,0)\), point \(A(0, - 4)\) and \(A'(0,-2)\).

Step2: Determine the scale factor

For the \(y\) - coordinate of \(B\) and \(B'\): if \(y = 4\) (for \(B\)) and \(y'=2\) (for \(B'\)), and for the \(x\) - coordinate of \(C\) and \(C'\): if \(x = 4\) (for \(C\)) and \(x'=2\) (for \(C'\)). The scale factor \(k\) can be found by the formula \(k=\frac{\text{image coordinate}}{\text{original coordinate}}\). For example, using \(B\) and \(B'\), \(k=\frac{2}{4}=\frac{1}{2}\), using \(C\) and \(C'\), \(k = \frac{2}{4}=\frac{1}{2}\).

Step3: Write the coordinate rule

The general rule for a dilation centered at the origin \((0,0)\) is \((x,y)\to(kx,ky)\). Since \(k = \frac{1}{2}\), the rule is \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)

Answer:

\((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)