Question

the circle shown below is centered at (0, 0) and passes through point p located at (4, 0).
the circle is dilated with the center of dilation at the origin and a scale factor of 0.5 and then translated up 3 units. what are the coordinates of the image of point p after this transformation?
a. (2, 3)
b. (3, 2)
c. (2, 3.5)
d. (8, 6)

Explanation:

Step1: Apply dilation

When a point \((x,y)\) is dilated with a scale factor \(k\) centered at the origin \((0,0)\), the new coordinates are \((kx,ky)\).
Given point \(P(4,0)\) and \(k = 0.5\), we have \(x'=0.5\times4 = 2\) and \(y'=0.5\times0=0\). So the point after dilation is \((2,0)\).

Step2: Apply translation

When a point \((x,y)\) is translated up \(3\) units, the rule is \((x,y + 3)\).
For the point \((2,0)\) after dilation, using the translation rule: \(x''=2\) and \(y''=0 + 3=3\).

Answer:

A. \((2,3)\)