Question

quadrilateral mnop was transformed to create quadrilateral efgh. determine if quadrilateral mnop is similar to quadrilateral efgh. a. quadrilateral mnop is not similar to quadrilateral efgh because a dilation centered at the origin by a scale factor of 4 maps quadrilateral mnop to quadrilateral efgh. b. quadrilateral mnop is similar to quadrilateral efgh because a dilation centered at the origin by a scale factor of 3 maps quadrilateral mnop to quadrilateral efgh. c. quadrilateral mnop is not similar to quadrilateral efgh because a dilation centered at the origin by a scale factor of 3 maps quadrilateral mnop to quadrilateral efgh. d. quadrilateral mnop is similar to quadrilateral efgh because a dilation centered at the origin by a scale factor of 4 maps quadrilateral mnop to quadrilateral efgh.

Explanation:

Step1: Find the length of corresponding sides

Let's assume the side \(MN\) in quadrilateral \(MNOP\). If \(M(- 2,0)\) and \(N(2,0)\), then the length of \(MN=\vert2 - (-2)\vert = 4\). In quadrilateral \(EFGH\), if \(E(-6,0)\) and \(F(6,0)\), then the length of \(EF=\vert6-(-6)\vert = 12\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\). So, \(k = \frac{EF}{MN}=\frac{12}{4}=3\). But for similar figures, all corresponding sides must be in proportion and the transformation should be a dilation (a similarity transformation). However, if we check the vertical sides (for example, if we assume a vertical side in \(MNOP\) has length \(1\) unit (from \(y = 0\) to \(y=-1\) for a small part of the quadrilateral near the \(x\) - axis) and in \(EFGH\) the corresponding vertical side has length \(3\) units (from \(y = 0\) to \(y = - 3\)). But if we consider the overall shape, a dilation is a similarity transformation. A dilation centered at the origin with scale factor \(k = 3\) maps \(MNOP\) to \(EFGH\)

Answer:

B. Quadrilateral \(MNOP\) is similar to quadrilateral \(EFGH\) because a dilation centered at the origin by a scale factor of \(3\) maps quadrilateral \(MNOP\) to quadrilateral \(EFGH\)