Questions: Look at the diagram below. The line NP and the line MQ will be dilated by a scale factor of 1/4 centered at M to create the line N'P' and the line M'Q, respectively. Select all of the true statements. - The line MQ' will be perpendicular to the line N. - The line N'P will be perpendicular to the line NP. - The line N'P will be parallel to the line NP, but they will not be the same line. - The line M'Q will be the same line as the line MQ.

Look at the diagram below.
The line NP and the line MQ will be dilated by a scale factor of 1/4 centered at M to create the line N'P' and the line M'Q, respectively.

Select all of the true statements.
- The line MQ' will be perpendicular to the line N.
- The line N'P will be perpendicular to the line NP.
- The line N'P will be parallel to the line NP, but they will not be the same line.
- The line M'Q will be the same line as the line MQ.

Transcript text: Look at the diagram below. $\overleftrightarrow{N P}$ and $\overleftrightarrow{M Q}$ will be dilated by a scale factor of $\frac{1}{4}$ centered at $M$ to create $\overleftrightarrow{N}{ }^{\prime} P^{\prime}$ and $\overleftrightarrow{M^{\prime} Q}$, respectively. Select all of the true statements. $\overleftrightarrow{M} Q^{\prime}$ will be perpendicular to $\overleftrightarrow{N}$. $\overleftrightarrow{N}{ }^{\prime} P$ will be perpendicular to $\overleftrightarrow{N P}$. $\overleftrightarrow{N ' P}$ will be parallel to $\overleftrightarrow{N P}$, but they will not be the same line. $\overleftrightarrow{M ' Q}$ will be the same line as $\overleftrightarrow{M Q}$.

Solution

Solution Steps

Step 1: Understanding the Problem

We are given a diagram with lines $ \overline{NP} $ and $ \overline{MQ} $ intersecting at point $ O $. These lines will be dilated by a scale factor of $ \frac{1}{4} $ centered at $ M $ to create $ \overline{N'P'} $ and $ \overline{M'Q'} $, respectively. We need to determine which statements about the new lines are true.

Step 2: Analyzing the Dilation

Dilation with a scale factor of $ \frac{1}{4} $ centered at $ M $ means that every point on $ \overline{NP} $ and $ \overline{MQ} $ will be moved closer to $ M $ by a factor of $ \frac{1}{4} $. This transformation will not change the angles between the lines, only their lengths.

Step 3: Evaluating the Statements

$ \overline{M'Q'} $ will be perpendicular to $ \overline{N'P'} $:
- Since dilation does not change the angles between lines, $ \overline{M'Q'} $ will remain perpendicular to $ \overline{N'P'} $.
$ \overline{N'P'} $ will be perpendicular to $ \overline{NP} $:
- $ \overline{N'P'} $ is a scaled version of $ \overline{NP} $ and will be parallel to $ \overline{NP} $, not perpendicular.
$ \overline{N'P'} $ will be parallel to $ \overline{NP} $, but they will not be the same line:
- This is true because dilation preserves parallelism but changes the length and position of the line.

Final Answer

$ \overline{M'Q'} $ will be perpendicular to $ \overline{N'P'} $.
$ \overline{N'P'} $ will be parallel to $ \overline{NP} $, but they will not be the same line.

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