# Characterize Points on a Perpendicular Bisector

### New York State Common Core Math Geometry, Module 1, Lesson 17

Worksheets for Geometry, Module 1, Lesson 17

Student Outcomes

• Students understand that any point on a line of reflection is equidistant from any pair of pre-image and image points in a reflection.

Characterize Points on a Perpendicular Bisector

Classwork

Opening Exercise

In Lesson 3, you bisected angles, including straight angles. You related the bisection of straight angles in Lesson 3 to the construction of perpendicular bisectors in Lesson 4. Review the process of constructing a perpendicular bisector with the segment below. Then complete the definition of perpendicular lines below your construction.

Use the compass and straightedge construction from Lesson 4.

Two lines are perpendicular if they ______, and if any of the angles formed by the intersection of the lines is a _____ angle. Two segments are perpendicular if the lines containing them are _______.

Discussion

The line you constructed in the Opening Exercise is called the perpendicular bisector of the segment. As you learned in Lesson 14, the perpendicular bisector is also known as the line of reflection of the segment. With a line of reflection, any point on one side of the line (pre-image) is the same distance from the line as its image on the opposite side of the line.

Example 1

Is it possible to find or construct a line of reflection that is not a perpendicular bisector of a segment connecting a point on the pre-image to its image? Try to locate a line of reflection between the two figures to the right without constructing any perpendicular bisectors.

Dicussion

Why were your attempts impossible? Look back at the definition of reflection from Lesson 14.

For a line 𝑙 in the plane, a reflection across 𝑙 is the transformation 𝑟𝑙 of the plane defined as follows:

1. For any point 𝑃 on the line 𝑙, 𝑟𝑙(𝑃) = 𝑃, and
2. For any point 𝑃 not on 𝑙, 𝑟𝑙(𝑃) is the point 𝑄 so that 𝑙 is the perpendicular bisector of the segment 𝑃𝑄.

The key lies in the use of the term perpendicular bisector. For a point 𝑃 not on 𝑙, explain how to construct the point 𝑄 so that 𝑙 is the perpendicular bisector of the segment 𝑃𝑄.

Now, let’s think about the problem from another perspective. We have determined that any point on the pre-image figure is the same distance from the line of reflection as its image. Therefore, the two points are equidistant from the point at which the line of reflection (perpendicular bisector) intersects the segment connecting the pre-image point to its image. What about other points on the perpendicular bisector? Are they also equidistant from the pre-image and image points? Let’s investigate.

Example 2

Using the same figure from the previous investigation, but with the line of reflection, is it possible to conclude that any point on the perpendicular bisector is equidistant from any pair of pre-image and image points? For example, is 𝐺𝑃 = 𝐻𝑃 in the figure? The point 𝑃 is clearly not on the segment connecting the pre-image point 𝐺 to its image 𝐻. How can you be certain that 𝐺𝑃 = 𝐻𝑃? If 𝑟 is the reflection, then 𝑟(𝐺) = 𝐻 and 𝑟(𝑃) = 𝑃. Since 𝑟 preserves distances, 𝐺𝑃 = 𝐻P

Discussion

We have explored perpendicular bisectors as they relate to reflections and have determined that they are essential to reflections. Are perpendicular lines, specifically, perpendicular bisectors, essential to the other two types of rigid motions: rotations and translations? Translations involve constructing parallel lines (which can certainly be done by constructing perpendiculars but are not essential to constructing parallels). However, perpendicular bisectors play an important role in rotations. In Lesson 13, we found that the intersection of the perpendicular bisectors of two segments connecting pairs of pre-image to image points determined the center of rotation.

Example 3

Find the center of rotation for the transformation below. How are perpendicular bisectors a major part of finding the center of rotation? Why are they essential?

As you explore this figure, also note another feature of rotations. As with all rigid motions, rotations preserve distance. A transformation is said to be distance-preserving (or length-preserving) if the distance between the images of two points is always equal to the distance between the original two points. Which of the statements below is true of the distances in the figure? Justify your response

Exercises

In each pre-image/image combination below, (a) identify the type of transformation; (b) state whether perpendicular bisectors play a role in constructing the transformation and, if so, what role; and (c) cite an illustration of the distancepreserving characteristic of the transformation (e.g., identify two congruent segments from the pre-image to the image). For the last requirement, you have to label vertices on the pre-image and image.

1. In the figure to the right, 𝐺𝐻 is a segment of reflection. State and justify two conclusions about distances in this figure. At least one of your statements should refer to perpendicular bisectors.

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