Question

problem 2
show that the figures are similar by describing a sequence of
transformations that moves the shaded pre - image onto the
unshaded image.

Explanation:

Step1: Analyze the transformation type

First, observe the orientation of the two rectangles. The orientation is the same, so there is no rotation. Then, check the size. The larger rectangle (un - shaded) is a scaled - up version of the smaller (shaded) rectangle. Also, we need to check the position.

Step2: Determine the translation

Count the number of units the shaded rectangle needs to move horizontally and vertically to align with the un - shaded rectangle. Let's assume the bottom - left corner of the shaded rectangle is at a certain grid point and the bottom - left corner of the un - shaded rectangle is at another.

Step3: Determine the dilation

Count the length and width of the shaded and un - shaded rectangles. Suppose the length of the shaded rectangle \(l_1 = 3\) units and the length of the un - shaded rectangle \(l_2=6\) units. The scale factor \(k=\frac{l_2}{l_1}\).

Let's assume the shaded rectangle (pre - image) has length \(l = 3\) units and width \(w = 2\) units, and the un - shaded rectangle (image) has length \(L = 6\) units and width \(W = 4\) units.

1. Translation:

• Suppose we consider the movement of a vertex (say the top - left vertex). Count the number of units to the left (or right) and up (or down). If we assume the shaded rectangle needs to move \(x\) units to the left and \(y\) units down (or vice - versa depending on the grid). Let's say we translate the shaded rectangle \(4\) units to the left and \(2\) units down.

1. Dilation:

• The scale factor \(k\) for dilation. For the length, if \(l_{shaded}=3\) and \(l_{un - shaded}=6\), then \(k = 2\). Using the formula for dilation \((x,y)\to(kx,ky)\)

Answer:

First, translate the shaded rectangle (pre - image) \(4\) units to the left and \(2\) units down. Then, dilate the translated rectangle by a scale factor of \(2\) with respect to the origin (or a fixed center of dilation). So the sequence of transformations is a translation followed by a dilation.