Question

figure pqrst was changed to create figure uvwxy. determine if figure pqrst is similar to figure uvwxy. a. figure pqrst is not similar to figure uvwxy because geometric stretch (x,y) to (0.25x,0.5y) maps figure pqrst to figure uvwxy. b. figure pqrst is similar to figure uvwxy because dilation (x,y) to (0.5x,0.5y) maps figure pqrst to figure uvwxy. c. figure pqrst is similar to figure uvwxy because dilation (x,y) to (0.25x,0.25y) maps figure pqrst to figure uvwxy. d. figure pqrst is not similar to figure uvwxy because geometric stretch (x,y) to (0.5x,0.25y) maps figure pqrst to figure uvwxy.

Explanation:

Step1: Understand similarity and dilation

Similar figures can be obtained by dilation (a transformation that enlarges or reduces a figure by a scale factor). A geometric stretch (non - uniform scaling in \(x\) and \(y\) directions) does not preserve similarity (since angles may change).

Step2: Check the scale factor

Let's assume a point. For example, if we consider a horizontal and vertical distance. Suppose in figure \(PQRST\), if we take a horizontal segment of length \(l_x\) and a vertical segment of length \(l_y\).
If we use a dilation \((x,y)\to(0.5x,0.5y)\):

• For a point \((x_1,y_1)\) in \(PQRST\), the image point in \(UVWXY\) is \((0.5x_1,0.5y_1)\). This is a uniform scaling (same scale factor \(k = 0.5\) for \(x\) and \(y\) directions).

A dilation \((x,y)\to(0.5x,0.5y)\) preserves the shape (ratios of side lengths and angles). A geometric stretch (e.g., \((x,y)\to(0.25x,0.5y)\) or \((x,y)\to(0.5x,0.25y)\)) changes the shape (ratios of side lengths in \(x\) and \(y\) directions are not the same).
A dilation \((x,y)\to(0.25x,0.25y)\) would make the figure too small (if we check the position of the vertices, for example, if \(Q\) is at \((2,6)\) in \(PQRST\), with \((x,y)\to(0.25x,0.25y)\) its image would be \((0.5,1.5)\) which is not the case for \(V\) (approximate \(x = 2,y = 3\)).

Answer:

B. Figure \(PQRST\) is similar to figure \(UVWXY\) because dilation \((x,y)\) to \((0.5x,0.5y)\) maps figure \(PQRST\) to figure \(UVWXY\).