Question

the points a and a are depicted on the graph below. the point a is obtained by dilating a about the origin. what is the scale factor of this dilation?

Explanation:

Step1: Find coordinates of \(A\) and \(A'\)

From the graph, \(A=(6, - 4)\) and \(A'=(4,-\frac{8}{3})\)

Step2: Use the dilation formula

If a point \(P(x,y)\) is dilated about the origin with scale factor \(k\) to get \(P'(x',y')\), then \(x' = kx\) and \(y'=ky\). Using the \(x -\)coordinates: \(k=\frac{x'}{x}\). Substitute \(x = 6\) and \(x'=4\), so \(k=\frac{4}{6}=\frac{2}{3}\). Check with \(y -\)coordinates: \(y=-4\), \(y'=-\frac{8}{3}\), and \(\frac{y'}{y}=\frac{-\frac{8}{3}}{-4}=\frac{2}{3}\)

Answer:

\(\frac{2}{3}\)