# Construct a Perpendicular Bisector

### New York State Common Core Math Geometry, Module 1, Lesson 4

Worksheets for Geometry, Module 1, Lesson 4

Student Outcomes

• Students learn to construct a perpendicular bisector and about the relationship between symmetry with respect to a line and a perpendicular bisector.

Construct a Perpendicular Bisector

Classwork

Opening Exercise

Choose one method below to check your Problem Set:

• Trace your copied angles and bisectors onto patty paper; then, fold the paper along the bisector you constructed. Did one ray exactly overlap the other?
• Work with your partner. Hold one partnerโs work over anotherโs. Did your angles and bisectors coincide perfectly?

Use the following rubric to evaluate your Problem Set:

Discussion

In Lesson 3, we studied how to construct an angle bisector. We know we can verify the construction by folding an angle along the bisector. A correct construction means that one half of the original angle coincides exactly with the other half so that each point of one ray of the angle maps onto a corresponding point on the other ray of the angle.

We now extend this observation. Imagine a segment that joins any pair of points that map onto each other when the original angle is folded along the bisector. The figure to the right illustrates two such segments.

Let us examine one of the two segments, ๐ธ๐บ. When the angle is folded along ๐ด๐ฝ, ๐ธ coincides with ๐บ. In fact, folding the angle demonstrates that ๐ธ is the same distance from ๐น as ๐บ is from ๐น; ๐ธ๐น = ๐น๐บ. The point that separates these equal halves of ๐ธ๐บ is ๐น, which is, in fact, the midpoint of the segment and lies on the bisector ๐ด๐ฝ. We can make this case for any segment that falls under the conditions above.

By using geometry facts we acquired in earlier school years, we can also show that the angles formed by the segment and the angle bisector are right angles. Again, by folding, we can show that โ ๐ธ๐น๐ฝ and โ ๐บ๐น๐ฝ coincide and must have the same measure. The two angles also lie on a straight line, which means they sum to 180ยฐ. Since they are equal in measure and sum to 180ยฐ, they each have a measure of 90ยฐ.

These arguments lead to a remark about symmetry with respect to a line and the definition of a perpendicular bisector. Two points are symmetric with respect to a line ๐ if and only if ๐ is the perpendicular bisector of the segment that joins the two points. A perpendicular bisector of a segment passes through the ________ of the segment and forms _________ with the segment.

We now investigate how to construct a perpendicular bisector of a line segment using a compass and a straightedge. Using what you know about the construction of an angle bisector, experiment with your construction tools and the following line segment to establish the steps that determine this construction.

Precisely describe the steps you took to bisect the segment.

Mathematical Modeling Exercise

You know how to construct the perpendicular bisector of a segment. Now, investigate how to construct a perpendicular to a line โ from a point ๐ด not on โ. Think about how you have used circles in constructions so far and why the perpendicular bisector construction works the way it does. The first step of the instructions has been provided for you. Discover the construction, and write the remaining steps.

Step 1. Draw circle ๐ด: center ๐ด with radius so that circle ๐ด intersects line โ in two points.

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