Perpendicular Bisector Theorem (Proof, Converse, Examples, & Video) (2024)

Perpendicular Bisector Theorem (Proof, Converse, & Examples)

Perpendicular

All good learning begins with vocabulary, so we will focus on the two important words of the theorem.Perpendicularmeans two line segments, rays, lines or any combination of those that meet at right angles. A line is perpendicular if it intersects another line and creates right angles.

Bisector

Abisectoris an object (a line, a ray, or line segment) that cuts another object (an angle, a line segment) into two equal parts. A bisector cannot bisect a line, because by definition a line is infinite.

Perpendicular bisector

Putting the two meanings together, we get the concept of aperpendicular bisector, a line, ray or line segment that bisects an angle or line segment at a right angle.

Before you get all bothered about it being a perpendicular bisector of anangle, consider: what is the measure of a straight angle?180°180°; that means a line dividing that angle into two equal parts and forming two right angles is a perpendicular bisector of the angle.

Perpendicular bisector theorem

Okay, we laid the groundwork. So putting everything together, what does thePerpendicular Bisector Theoremsay?

How does it work?

Suppose you have a big, square plot of land,1,000meterson a side. You built a humdinger of a radio tower,300metershigh, right smack in the middle of your land. You plan to broadcast rock music day and night.

Anyway, that location for your radio tower means you have500metersof land to the left, and500metersof land to the right. Your radio tower is a perpendicular bisector of the length of your land.

You need to reinforce the tower with wires to keep it from tipping over in high winds. Those are called guy wires. How long should a guy wire from the top down to the land be, on each side?

Because you constructed a perpendicular bisector, you do not need to measure on each side. One measurement, which you can calculate using geometry, is enough. Use thePythagorean Theoremfor right triangles:

Your tower is300meters. You can go out500metersto anchor the wire's end. The tower meets your land at90°. So:

You need guy wires a whopping583.095meterslong to run from the top of the tower to the edge of your land. You repeat the operation at the200meterheight, and the100meterheight.

For every height you choose, you will cut guy wires of identical lengths for the left and right side of your radio tower, because the tower is the perpendicular bisector of your land.

Proving the perpendicular bisector theorem

Behold the awesome power of the two words, "perpendicular bisector," because with only a line segment,HM, and its perpendicular bisector,WA, we can prove this theorem.

We are given line segmentHMand we have bisected it (divided it exactly in two) by a lineWA. That line bisectedHMat90°because it is a given. This means, if we run a line segment fromPointWtoPointH, we can create right triangleWHA, and another line segmentWMcreates right triangleWAM.

What do we have now? We have two right triangles,WHAandWAM, sharing sideWA, with all these congruences:

1. WA ≅ WA (by the reflexive property)

2. ∠WAH ≅ ∠WAM (90° angles; given)

3. HA ≅ AM (bisector; given)

What does that look like? We hope you saidSide Angle Side, because that is exactly what it is.

That means sidesWHandWMare congruent, because CPCTC (corresponding parts of congruent triangles are congruent).WHAM!Proven!

Practice proof

You can tackle the theorem yourself now. You will either sink or swim on this one. Here is a line segment,WM. We construct a perpendicular bisector,SI.

How can you prove thatSW≅SM? Do you know what to do?

1. Construct line segments SW and SM.

2. You now have what? Two right triangles, SWI and SIM. They have right angles, ∠SIW and ∠SIM.

3. Identify WI and IM as congruent, because they are the two parts of line segment WM that were bisected by SI.

4. Identify SI as congruent to itself (by the reflexive property).

What does that give you? Two congruent sides and an included angle, which is what postulate? TheSAS Postulate, of course! Therefore, line segmentSW≅SM.

So, did you sink orSWIM?

Converse of the perpendicular bisector theorem

Notice that the theorem is constructed as an "if, then" statement. That immediately suggests you can write the converse of it, by switching the parts:

We can show this, too. Construct a line segmentHD. Place a random point above it (but still somewhere betweenPointsHandD) and call itPointT.

IfPointTis the same distance fromPointsHandD, this converse statement says it must lie on the perpendicular bisector ofHD.

You can prove or disprove this by dropping a perpendicular line fromPointTthrough line segmentHD. Where your perpendicular line crossesHD, call itPointU.

IfPointTisthe same distance fromPointsHandD, thenHU≅UD. IfPointTisnotthe same distance fromPointsHandD, thenHU≆UD.

You can go through the steps of creating two right triangles,△THUand△TUDand proving angles and sides congruent (or not congruent), the same as with the original theorem.

You would identify the right angles, the congruent sides along the original line segmentHD, and the reflexive congruent sideTU. When you got to a pair of corresponding sides that were not congruent, then you would knowPointTwas not on the perpendicular bisector.

Only points lying on the perpendicular bisector will be equidistant from the endpoints of the line segment. Everything else lands with aTHUD.

Lesson summary

After you worked your way through all the angles, proofs and multimedia, you are now able to recall the Perpendicular Bisector Theorem and test the converse of the Theorem. You also got a refresher in what "perpendicular," "bisector," and "converse" mean.