Question

the pre-image abcd is dilated to become the image abcd. the origin is the center of dilation. what scale factor is used to create the dilation?

Explanation:

Step1: Identify coordinates of a point

Let's take point \( A \) and \( A' \). From the graph, assume \( A \) has coordinates \( (2, 4) \) and \( A' \) has coordinates \( (2, 8) \) (or other corresponding points, but let's check the side length). Alternatively, check the side length of the pre - image and image. Let's assume the pre - image square \( ABCD \) has a side length of \( s \) and the image square \( A'B'C'D' \) has a side length of \( s' \). If we consider the horizontal or vertical distance between two points. For example, if the pre - image has a side length of 2 units (from \( x = 2\) to \( x = 4\) for a side) and the image has a side length of 4 units (from \( x = 2\) to \( x = 6\))? Wait, maybe better to use the ratio of corresponding coordinates. Since the center of dilation is the origin, the scale factor \( k=\frac{\text{coordinate of image point}}{\text{coordinate of pre - image point}}\) for a point \((x,y)\) and its image \((kx,ky)\). Let's find a point, say \( A \) is at \((2,4)\) and \( A' \) is at \((2,8)\)? No, maybe \( A \) is at \((2,4)\) and \( A' \) is at \((2,8)\) is not correct. Wait, looking at the graph, the pre - image (small square) and the image (large square). Let's assume the pre - image has a side length of 2 (from \( y = 4\) to \( y = 6\) for a vertical side) and the image has a side length of 4 (from \( y = 4\) to \( y = 8\))? Wait, no, let's take the coordinates of \( A \) and \( A' \). Suppose \( A=(2,4) \) and \( A'=(2,8) \), then the scale factor \( k=\frac{y_{A'}}{y_{A}}=\frac{8}{4} = 2\). Or if \( A=(2,4) \) and \( A'=(2,8) \), the \( y \) - coordinate of \( A' \) is twice that of \( A \), so the scale factor is 2.

Step2: Verify with another point

Take point \( B \) and \( B' \). Suppose \( B=(4,4) \) and \( B'=(4,8) \), then \( \frac{y_{B'}}{y_{B}}=\frac{8}{4}=2 \). So the scale factor is 2.

Answer:

2