Question

write a rule to describe each transformation.
16)
a) dilation of $\frac{1}{2}$ about the origin
b) dilation of $\frac{1}{4}$ about the origin
c) dilation of 1.5 about the origin
d) dilation of 0.25 about the origin
17)
a) dilation of 2 about the origin
b) dilation of 2.5 about the origin
c) dilation of $\frac{3}{2}$ about the origin
d) dilation of 1.5 about the origin

Explanation:

Step1: Recall the formula for dilation

The formula for dilation about the origin is \((x,y)\to(kx,ky)\), where \(k\) is the scale factor.

Step2: Analyze problem 16

Assume a point on the larger figure and its corresponding point on the smaller figure. Let's say a vertex of the larger figure has coordinates \((x_1,y_1)\) and the corresponding vertex of the smaller figure has coordinates \((x_2,y_2)\). If we take a point (for example, if a side - length of the larger figure is \(4\) units and the corresponding side - length of the smaller figure is \(2\) units). Using the formula \(k=\frac{x_2}{x_1}=\frac{y_2}{y_1}\). If we assume \(x_1 = 4\) and \(x_2=2\), then \(k=\frac{1}{2}\)

Step3: Analyze problem 17

Assume a point on the smaller figure and its corresponding point on the larger figure. Let's say a vertex of the smaller figure has coordinates \((x_3,y_3)\) and the corresponding vertex of the larger figure has coordinates \((x_4,y_4)\). If we take a side - length of the smaller figure as \(2\) units and the corresponding side - length of the larger figure as \(3\) units. Using the formula \(k = \frac{x_4}{x_3}=\frac{y_4}{y_3}\). If \(x_3 = 2\) and \(x_4=3\), then \(k=\frac{3}{2}\)

Answer:

1. A. dilation of \(\frac{1}{2}\) about the origin
2. C. dilation of \(\frac{3}{2}\) about the origin