QR passes through point S and is perpendicular to TU at S. Suppose QR is dilated by a scale factor of 1/3 centered on point S. Which of the following statements is true for the dilated lines?

1. The dilated line segments intersect at S but are not perpendicular
2. The dilated line segments intersect at a point other than S
3.The dilated lines are perpendicular
4. the dilated lines are parallel

only the length of QR is changed, so it is still ⊥ to TU

To determine the answer, let's analyze the information given and consider the effects of the dilation.

We know that the line QR is perpendicular to TU at point S. This means that the angle formed at S between QR and TU is 90 degrees.

The dilation has a scale factor of 1/3, meaning that each point on QR will be moved to a new position that is 1/3 of the distance from S.

Now let's consider the possible outcomes:

1. The dilated line segments intersect at S but are not perpendicular:
If the dilated line segments intersect at S, they would have to be perpendicular to each other to maintain the characteristics of the original perpendicular lines. Therefore, this statement is not true.

2. The dilated line segments intersect at a point other than S:
If the dilated line segments intersect at a point other than S, it would mean that the lines are not parallel and hence not perpendicular to TU. Therefore, this statement is not true.

3. The dilated lines are perpendicular:
Since the original line QR is perpendicular to TU, when it is dilated by a scale factor of 1/3 centered on S, the resulting lines will still be perpendicular. Therefore, this statement is true.

4. The dilated lines are parallel:
If the dilated lines were parallel, it would mean that the original line QR was parallel to TU, which contradicts the given information that QR is perpendicular to TU. Therefore, this statement is not true.

In conclusion, the correct answer is:
3. The dilated lines are perpendicular.