Question

which statements are true about the parallelograms? choose three correct answers
the scale factor is $\frac{2}{5}$.
the image is smaller than the pre - image.
sides $mq$ and $mq$ both have the same slope, 1.
the length of side $mn$ is 2 units.
the length of side $mn$ is 5 units.

Explanation:

Step1: Calculate the scale factor

The length of \( MN = 2 \) units and the length of \( M'N'=5 \) units. The scale factor \( k=\frac{M'N'}{MN}=\frac{5}{2} \). So the first statement “The scale factor is \(\frac{2}{5}\)” is wrong.

Step2: Compare the size of image and pre - image

Since the scale factor \( k = \frac{5}{2}>1 \), the image \( M'N'P'Q'\) is larger than the pre - image \( MNPQ\). So the second statement “The image is smaller than the pre - image” is wrong.

Step3: Calculate the slope of \( MQ \) and \( M'Q' \)

Let \( M(x_1,y_1)\) and \( Q(x_2,y_2)\). Assume \( M(- 1,2)\), \( Q(-3,2)\), \( M'(-5,6)\), \( Q'(-9,6)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For \( MQ\): \(m_{MQ}=\frac{2 - 2}{-3+1}=0\) (using \(M(-1,2)\) and \(Q(-3,2)\)).
For \( M'Q'\): \(m_{M'Q'}=\frac{6 - 6}{-9 + 5}=0\) (using \(M'(-5,6)\) and \(Q'(-9,6)\)).
In general, for a dilation, corresponding sides are parallel. Parallel lines have the same slope.

Step4: Check the length of \( MN \) and \( M'N' \)

By counting the units on the coordinate - grid:
The length of \( MN\): If \( M(-1,4)\) and \( N(-1,2)\), then \(MN=\vert4 - 2\vert=2\) units.
The length of \( M'N'\): If \( M'(-5,10)\) and \( N'(-5,5)\), then \(M'N'=\vert10 - 5\vert = 5\) units.

Answer:

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