Question

which description of a figure and its image match the transformation described by $(x,y)\to(x,3y)$?
a the figure and its image are similar and the image has been translated 3 units up.
b the figure and its image are similar and the image has been translated 3 units to the right.
c the figure and its image are neither similar nor congruent, and the image has been stretched horizontally.
d the figure and its image are neither similar nor congruent, and the image has been stretched vertically.

Explanation:

• For a transformation \((x,y)\to(x,3y)\), the \(x -\)coordinate remains the same. So there is no horizontal translation (left - right movement).
• The \(y -\)coordinate is multiplied by \(3\). This is a vertical stretch.
• Congruent figures have the same shape and size. Similar figures have the same shape but can have different sizes. When we stretch a figure vertically (by a factor other than \(1\)), the ratios of the corresponding sides (for non - vertical sides) change. For example, if we have a right - triangle with vertices \((0,0)\), \((0,1)\), \((1,0)\) and we apply the transformation \((x,y)\to(x,3y)\), the new vertices are \((0,0)\), \((0,3)\), \((1,0)\). The original triangle has side lengths \(1\), \(1\), \(\sqrt{2}\) and the new triangle has side lengths \(1\), \(3\), \(\sqrt{1 + 9}=\sqrt{10}\). The ratios of the non - vertical sides (e.g., if we consider the sides connecting \((0,0)\) to \((1,0)\) and \((0,0)\) to \((0,1)\) or \((0,0)\) to \((0,3)\)) are not in proportion for congruence or similarity (except in the case of a vertical line, but in general for a non - degenerate figure).

Answer:

D. The figure and its image are neither similar nor congruent, and the image has been stretched vertically.