Question

describe a similarity transformation that maps the black preimage to the dashed image

Explanation:

Step1: Determine the scale factor

• Count the distances between corresponding points. For example, if we consider a side length in the pre - image and the image.
• Let's assume a side in the pre - image. Suppose we look at the horizontal or vertical segments. If we assume the pre - image has a certain length and the image has a length that is a multiple of it.
• The scale factor \(k\) can be found by comparing the lengths of corresponding sides. If we assume a vertical or horizontal side. Let's say the pre - image has a side of length \(l_1\) and the image has a side of length \(l_2\). The scale factor \(k=\frac{l_2}{l_1}\). From the graph, if we assume a side in the pre - image (smaller figure) and the image (larger figure), we can see that the scale factor \(k = 3\) (by counting the grid units for corresponding sides).

Step2: Determine the translation

• After dilation (scaling), we need to check the position.
• Let's assume we first dilate the pre - image with a scale factor of \(3\) centered at the origin. Then we need to translate.
• If we consider the center of dilation (origin) and then check the position of the dilated figure compared to the image.
• The translation is \( (x,y)\to(x + 3,y-12)\) (by comparing the position of a key point like \(A\) in the pre - image \((-1, - 2)\) (assuming the origin is at the intersection of the axes in the standard way, but actually, if we consider the grid, for the pre - image: assume \(A=( - 1,-2)\), \(B=(-3,-2)\), \(D=(-1, - 1)\), \(C=(-3,-1)\) and for the image \(A'=(2, - 14)\), \(B'=(4,-14)\), \(D'=(2,-13)\), \(C'=(4,-13)\)). After dilation with scale factor \(3\) centered at the origin \((x,y)\to(3x,3y)\), the points become \(A_d=(-3,-6)\), \(B_d=(-9,-6)\), \(D_d=(-3,-3)\), \(C_d=(-9,-3)\). Then the translation \((x,y)\to(x + 6,y - 8)\) (another way: by vector analysis of the movement of a single point. For example, take point \(A\) in pre - image. If we assume a dilation of scale factor \(3\) and then a translation. Let the pre - image point \(A=(x_1,y_1)\) and image point \(A'=(x_2,y_2)\). If we first do a dilation \((x,y)\to(3x,3y)\) and then a translation \((x,y)\to(x + a,y + b)\). Solving \(3x_1+a=x_2\) and \(3y_1 + b=y_2\). If \(x_1=-1,y_1=-2,x_2 = 2,y_2=-14\), then \(a=2-3\times(-1)=5\) and \(b=-14 - 3\times(-2)=- 8\). But a more accurate way by counting grid units:
• First, dilate the pre - image by a scale factor of \(3\) (enlarge the figure by a factor of \(3\)). Then translate the dilated figure \(6\) units to the right and \(8\) units down.

Answer:

Dilate the pre - image by a scale factor of \(3\) centered at the origin and then translate \(6\) units to the right and \(8\) units down.