Question

for the figure below, do a dilation centered at the origin with a scale factor of $\frac{1}{2}$. then, give the endpoints for both the original figure and the final figure.

Explanation:

1. First, assume the endpoints of the original - line segment:

• Let's say the endpoints of the original line - segment are \((x_1,y_1)\) and \((x_2,y_2)\). From the graph, if we assume the left - hand endpoint of the original line segment is \((1,2)\) and the right - hand endpoint is \((3,6)\).

Step1: Recall the dilation formula

The formula for dilation centered at the origin with a scale factor \(k\) is \((x,y)\to(kx,ky)\). Here \(k = \frac{1}{2}\).

Step2: Calculate the endpoints of the dilated figure for the left - hand endpoint

For the left - hand endpoint \((x_1,y_1)=(1,2)\), after dilation, the new coordinates \((x_1',y_1')\) are given by \(x_1'=\frac{1}{2}\times1=\frac{1}{2}\), \(y_1'=\frac{1}{2}\times2 = 1\).

Step3: Calculate the endpoints of the dilated figure for the right - hand endpoint

For the right - hand endpoint \((x_2,y_2)=(3,6)\), after dilation, the new coordinates \((x_2',y_2')\) are given by \(x_2'=\frac{1}{2}\times3=\frac{3}{2}\), \(y_2'=\frac{1}{2}\times6 = 3\).

Answer:

Endpoints of original figure: Left: \((1,2)\) Right: \((3,6)\)
Endpoints of final figure: Left: \((\frac{1}{2},1)\) Right: \((\frac{3}{2},3)\)