What is a Perpendicular Bisector?
In geometry, a perpendicular bisector is a line segment that intersects a given line segment at its midpoint. The term "bisector" refers to the fact that the line segment cuts the given line segment into two equal halves. A perpendicular bisector is simply a perpendicular line segment that also happens to be a bisector. In this blog post, we'll explore what perpendicular bisectors are and how they can be used in mathematical proofs.
How to Construct a Perpendicular Bisector
There are many ways to construct a perpendicular bisector. One way is to use a compass. First, find the midpoint of the given line segment. Then, place the point of the compass at the midpoint and draw an arc that intersects both ends of the given line segment. Next, without changing the width of the compass, place the point of the compass at one end of the given line segment and draw another arc that intersects the first arc. Finally, draw a straight line through the point of intersection of the two arcs to create your perpendicular bisector.
Another way to construct a perpendicular bisector is with a straightedge (ruler). First, find the midpoint of the given line segment as before. Then, place the straightedge so that it intersects the midpoint and extend it on both sides of the midpoint an equal distance. Finally, draw a straight line through the point of intersection of the two lines to create your perpendicular bisector.
Uses for Perpendicular Bisectors in Mathematical Proofs
One common use for perpendicular bisectors is in proving triangles congruent by ASA (angle-side-angle) or AAS (angle-angle-side). ASA states that if two angles and one side of one triangle are congruent to two angles and one side of another triangle, then those triangles are congruent. AAS states that if two angles and a corresponding non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle, then those triangles are congruent. In order to use either ASA or AAS to prove triangles congruent, you must first construct perpendicular bisectors for each triangle.
Conclusion:
Perpendicular bisectors are important tools in geometry that can be used for various purposes, such as constructing mathematical proofs. In this blog post, we explored what perpendicular Bisectors are and how they can be constructed using either a compass or ruler. We also discussed how perpendicular bisectors can be used in mathematical proofs involving ASA and AAS congruence criteria.
FAQ
What is a perpendicular bisector in geometry?
In geometry, a perpendicular bisector is a line segment that intersects a given line segment at its midpoint. The term "bisector" refers to the fact that the line segment cuts the given line segment into two equal halves. A perpendicular bisector is simply a perpendicular line segment that also happens to be a bisector.
What is an example of a perpendicular bisector?
One common use for perpendicular bisectors is in proving triangles congruent by ASA (angle-side-angle) or AAS (angle-angle-side). ASA states that if two angles and one side of one triangle are congruent to two angles and one side of another triangle, then those triangles are congruent. AAS states that if two angles and a corresponding non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle, then those triangles are congruent. In order to use either ASA or AAS to prove triangles congruent, you must first construct perpendicular bisectors for each triangle.
Where is the perpendicular bisector?
There is no definitive answer to this question since the perpendicular bisector can be constructed in many different ways. However, one common method is to find the midpoint of the given line segment and then draw a perpendicular line segment through that point.
What are the 3 steps in constructing a perpendicular bisector?
There are three steps in constructing a perpendicular bisector:
1) Find the midpoint of the given line segment.
2) Place the point of the compass at the midpoint and draw an arc that intersects both ends of the given line segment.
3) Draw a straight line through the point of intersection of the two lines to create your perpendicular bisector.