Question

this review is three of five for the upcoming district mid - term exam. the exam will cover units 1 - 6. please complete your assignments for comprehension. (ask questions, if necessary)
geometry
midterm review

1. select the transformations that will create an image that is similar to the preimage?

a $(x,y)\to(2x,2y)$
b $(x,y)\to(y, - x)$
c $(x,y)\to(7x,5y)$
d $(x,y)\to(x - 4,y + 6)$
e $(x,y)\to(x,y - 3)$
f $(x,y)\to(x,2y)$

Explanation:

Step1: Recall the definition of similar figures

Similar figures have the same shape but not necessarily the same size. Transformations that preserve the shape (i.e., ratios of side - lengths) are similarity transformations.

• Translation:

A translation \((x,y)\to(x - 4,y + 6)\) (Option D) and \((x,y)\to(x,y - 3)\) (Option E) are rigid - body movements. They do not change the size or shape of the figure. Since similar figures can have different sizes (but same shape), translations are part of similarity transformations (they are a special case of similarity transformations with a scale factor of \(k = 1\)).

• Rotation:

A rotation \((x,y)\to(y,-x)\) (Option B) is a rigid - body transformation. Rotations preserve the shape and size of the figure. Since a rotation with a scale factor \(k = 1\) is a similarity transformation (as it preserves the shape).

• Dilation:

A dilation \((x,y)\to(2x,2y)\) (Option A) changes the size of the figure by a scale factor \(k = 2\). The ratios of the side - lengths of the pre - image and the image are equal. For a general point \((x_1,y_1)\) and its image \((2x_1,2y_1)\) and another point \((x_2,y_2)\) and its image \((2x_2,2y_2)\), the distance between \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and the distance between \((2x_1,2y_1)\) and \((2x_2,2y_2)\) is \(D=\sqrt{(2x_2 - 2x_1)^2+(2y_2 - 2y_1)^2}=2\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). The ratio of the distances \(D/d = 2\).

• Non - uniform scaling:

A non - uniform scaling \((x,y)\to(7x,5y)\) (Option C) and \((x,y)\to(x,2y)\) (Option F) does not preserve the ratios of the side - lengths. For example, consider a square with vertices \((0,0)\), \((1,0)\), \((1,1)\) and \((0,1)\). After the transformation \((x,y)\to(7x,5y)\), the vertices become \((0,0)\), \((7,0)\), \((7,5)\) and \((0,5)\). The original side - length of the square (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)) for adjacent vertices \((0,0)\) and \((1,0)\) is \(1\), and for \((1,0)\) and \((1,1)\) is \(1\). After transformation, the distance between \((0,0)\) and \((7,0)\) is \(7\) and between \((7,0)\) and \((7,5)\) is \(5\). The ratios of the side - lengths are not equal. Similarly, for \((x,y)\to(x,2y)\), a square will be transformed into a rectangle (non - similar).

Answer:

A. \((x,y)\to(2x,2y)\), B. \((x,y)\to(y,-x)\), D. \((x,y)\to(x - 4,y + 6)\), E. \((x,y)\to(x,y - 3)\)