## Square LMNO is shown below with line PQ passing through the center:If square LMNO is dilated by a scale factor of two about the center of the square to create square L’M’N’O’, dilated line P’Q’ will be parallel to line PQ and pass through point Nshift two units to the rightbe perpendicular to line PQ and pass through point Lcontain the points P and Q

### In order to understand the properties of the dilated line P’Q’, we first need to understand what dilation is and how it affects geometrical figures

In order to understand the properties of the dilated line P’Q’, we first need to understand what dilation is and how it affects geometrical figures.

Dilation is a transformation that changes the size of a figure without altering its shape. It involves scaling the figure up or down by a certain factor, called the scale factor. In this case, the square LMNO is being dilated by a scale factor of two, meaning that every side of the square becomes twice as long in the dilated square L’M’N’O’.

Now let’s analyze each of the statements about the dilated line P’Q’:

1. P’Q’ will be parallel to line PQ and pass through point N:

When a figure is dilated, the relationship between parallel lines is preserved. Since line PQ passes through the center of the square, it remains unchanged during dilation. Thus, P’Q’ will also be parallel to line PQ. Additionally, since point N is on line PQ, it will also lie on the dilated line P’Q’.

2. P’Q’ will shift two units to the right:

Dilation does not involve shifting a figure horizontally or vertically. Instead, it purely focuses on scaling the figure. The position of line PQ will remain the same after dilation, so P’Q’ will also stay in the same vertical position as PQ.

3. P’Q’ will be perpendicular to line PQ and pass through point L:

Perpendicularity is not preserved during dilation. As mentioned earlier, dilation only scales the figure without changing its shape or angles. Therefore, the dilated line P’Q’ will not necessarily be perpendicular to line PQ. It will maintain the same orientation as line PQ.

4. P’Q’ will contain the points P and Q:

Since line PQ passes through the center of the square, its endpoints are coincident with the vertices of the square. During dilation, the center remains fixed, so P’ and Q’ will coincide with the position of P and Q in the dilated square. Therefore, the dilated line P’Q’ will indeed contain the points P and Q.

Based on these observations, we can conclude that the correct statements about the dilated line P’Q’ are:

– P’Q’ will be parallel to line PQ and pass through point N.

– P’Q’ will contain the points P and Q.

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