Question

the graph shows quadrilateral abcd and the center of dilation at the origin, (0,0). the image of point b after dilation is labeled as b and is located at (6,2). explain how you know the scale factor. then describe how the shape changes in size and position after the dilation.

Explanation:

Step1: Calculate the scale factor

The formula for dilation with center at the origin \((0,0)\) is \((x,y)\to(kx,ky)\), where \(k\) is the scale factor.
For point \(B(3,1)\) and its image \(B'(6,2)\), we can find \(k\) by using the \(x -\)coordinates (or \(y -\)coordinates). Using the \(x -\)coordinates: \(kx_{B}=x_{B'}\), so \(k\times3 = 6\), then \(k=\frac{6}{3}=2\). Using the \(y -\)coordinates: \(ky_{B}=y_{B'}\), so \(k\times1 = 2\), then \(k = 2\).

Step2: Describe the change in size

Since \(k = 2>1\), the shape (quadrilateral \(A'B'C'D'\)) is larger than the original quadrilateral \(ABCD\). Each side length of the image is \(k = 2\) times the corresponding side length of the original figure. For example, if we consider a side \(AB\) with length \(d=\sqrt{(x_{A}-x_{B})^{2}+(y_{A}-y_{B})^{2}}=\sqrt{(- 2 - 3)^{2}+(3 - 1)^{2}}=\sqrt{25 + 4}=\sqrt{29}\), the length of \(A'B'\) is \(d'=\sqrt{(x_{A'}-x_{B'})^{2}+(y_{A'}-y_{B'})^{2}}\), where \(x_{A'}=-2\times2=-4\), \(y_{A'}=3\times2 = 6\), \(x_{B'}=6\), \(y_{B'}=2\), \(d'=\sqrt{(-4 - 6)^{2}+(6 - 2)^{2}}=\sqrt{100+16}=\sqrt{116}=2\sqrt{29}\)

Step3: Describe the change in position

Since the center of dilation is at the origin \((0,0)\), the position of the shape changes such that each point \((x,y)\) of quadrilateral \(ABCD\) is moved along the line joining the origin \((0,0)\) and the point \((x,y)\) to a new point \((kx,ky)\). The figure is expanded away from the origin.

Answer:

The scale factor \(k = 2\). The size of the shape (quadrilateral) doubles (each side length of the image is twice the corresponding side length of the original quadrilateral). The position of the shape changes as each point of the original quadrilateral \(ABCD\) is moved along the line connecting the point to the origin \((0,0)\) to a new point that is twice as far from the origin as the original point.