Construct a Perpendicular Bisector
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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, 𝐸𝐺. When the angle is folded along 𝐴𝐽, 𝐸 coincides with 𝐺. In fact, folding the angle demonstrates that 𝐸 is the same distance from 𝐹 as 𝐺 is from 𝐹; 𝐸𝐹 = 𝐹𝐺. The point that separates these equal halves of 𝐸𝐺 is 𝐹, which is, in fact, the midpoint of the segment and lies on the bisector 𝐴𝐽. 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 ∠𝐸𝐹𝐽 and ∠𝐺𝐹𝐽 coincide and must have the same measure. The two angles also lie on a straight line, which means they sum to 180°. Since they are equal in measure and sum to 180°, they each have a measure of 90°.
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 𝑙 if and only if 𝑙 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 ℓ from a point 𝐴 not on ℓ. 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 𝐴: center 𝐴 with radius so that circle 𝐴 intersects line ℓ in two points.
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