Question

consider the diagram.
part a
which point is the image of point ( r ) under a dilation with center ( o ) and scale factor ( \frac{1}{3} )?

Explanation:

Step1: Recall the property of dilation

Dilation with center \(O\) and scale factor \(k=\frac{1}{3}\) means that if a point \(P(x,y)\) has its image \(P'(x',y')\) under the dilation, then \(\overrightarrow{OP'}=k\overrightarrow{OP}\). In terms of position from the center \(O\), the image point is closer to \(O\) than the original point when \(0 < k<1\).

Step2: Analyze the positions of points

Since the scale factor \(k = \frac{1}{3}\), the image of point \(R\) should be a point that lies on the line segment \(OR\) and is \(\frac{1}{3}\) of the distance from \(O\) to \(R\). Looking at the diagram, point \(Q\) lies on the line - segment \(OR\) (assuming the points are collinear in the order \(O - Q - R\) in terms of the dilation concept) and the relative position of \(Q\) with respect to \(O\) and \(R\) satisfies the dilation property with scale factor \(\frac{1}{3}\).

Answer:

Point \(Q\)