Question

parallelogram mnpq was dilated to create parallelogram mnpq. which statements are true about the parallelograms? select three options. the length of side mn is 2 units. the length of side mn is 5 units. the image is smaller than the pre - image. sides mq and mq both have the same slope, 1. the scale factor is \frac{2}{5}.

Explanation:

Step1: Calculate the length of MN

Using the distance formula for horizontal line (since \(y -\)coordinates of \(M\) and \(N\) are same). If \(M=(2,- 2)\) and \(N=(4,-2)\), then \(MN=\vert4 - 2\vert=2\) units.

Step2: Calculate the length of M'N'

If \(M'=(4,-5)\) and \(N'=(9,-5)\), then \(M'N'=\vert9 - 4\vert = 5\) units.

Step3: Calculate the slope of MQ and M'Q'

For \(M=(2,-2)\) and \(Q=(1,-3)\), slope \(m_{MQ}=\frac{-3+2}{1 - 2}=\frac{-1}{-1}=1\). For \(M'=(4,-5)\) and \(Q'=(3,-6)\), slope \(m_{M'Q'}=\frac{-6 + 5}{3-4}=\frac{-1}{-1}=1\).

Step4: Check scale - factor and size comparison

Let the scale factor be \(k\). If we consider the length of corresponding sides \(MN = 2\) and \(M'N'=5\), then \(k=\frac{M'N'}{MN}=\frac{5}{2}\). Since \(k=\frac{5}{2}>1\), the image (\(M'N'P'Q'\)) is larger than the pre - image (\(MNPQ\)). So, the statements “The image is smaller than the pre - image” and “The scale factor is \(\frac{2}{5}\)” are false.

Answer:

• The length of side MN is 2 units.
• The length of side M'N' is 5 units.
• Sides MQ and M'Q' both have the same slope, 1.