Perpendicular Bisector theorem (Proof, Converse, & Examples)


All good learning starts with vocabulary, so us will emphasis on the two essential words the the theorem. Perpendicular method two line segments, rays, lines or any mix of those that accomplish at ideal angles. A line is perpendicular if it intersects one more line and creates ideal angles.

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A bisector is an item (a line, a ray, or line segment) the cuts one more object (an angle, a line segment) right into two equal parts. A bisector can not bisect a line, since by an interpretation a line is infinite.

Perpendicular Bisector

Putting the two definitions together, we acquire the ide of a perpendicular bisector, a line, beam or line segment the bisects an edge or heat segment at a ideal angle.

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

Perpendicular Bisector Theorem

Okay, we laid the groundwork. So putting every little thing together, what go the Perpendicular Bisector Theorem say?


How Does that Work?

Suppose you have a big, square plot that land, 1,000 meters on a side. You developed a humdinger that a radio tower, 300 meters high, appropriate smack in the middle of your land. You plan to broadcast rock music day and also night.

Anyway, that ar for your radio tower way you have 500 meters that land to the left, and 500 meters the land come the right. Your radio tower is a perpendicular bisector that the length of your land.


You have to reinforce the tower with wires to store it indigenous tipping end in high winds. Those are referred to as guy wires. Exactly how long have to a guy wire from the height down come the soil be, on each side?

Because you created a perpendicular bisector, you execute not should measure on each side. One measurement, i beg your pardon you can calculate utilizing geometry, is enough. Use the Pythagorean Theorem for appropriate triangles:

Your tower is 300 meters. You have the right to go out 500 meters to anchor the wire"s end. The tower meets your land in ~ 90°. So:

300 m2 + 500 m2 = c2

90,000 + 250,000 = c2

340,000 = c2

340,000 = c

583.095 m = c

You need male wires a lining 583.095 meters lengthy to operation from the top of the tower come the leaf of her land. Girlfriend repeat the operation at the 200 meter height, and the 100 meter height.

For every height you choose, friend will reduced guy wires of similar lengths for the left and right next of her radio tower, due to the fact that the tower is the perpendicular bisector of her land.

Proving the Perpendicular Bisector Theorem

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


We are provided line segment HM and also we have bisected it (divided it precisely in two) by a line WA. That line bisected HM at 90° due to the fact that it is a given. This means, if we operation a line segment indigenous Point W to Point H, we can develop right triangle WHA, and also another line segment WM creates ideal triangle WAM.

What perform we have actually now? We have actually two right triangles, WHA and also WAM, sharing side WA, with all this congruences:

WA ≅ WA (by the reflex property)∠WAH ≅ ∠WAM (90° angles; given)HA ≅ AM (bisector; given)

What does the look like? we hope you stated Side edge Side, due to the fact that that is precisely what that is.

That means sides WH and also WM room congruent, due to the fact that CPCTC (corresponding parts of congruent triangles space congruent). WHAM! Proven!

Practice Proof

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


How can you prove the SW ≅ SM? execute you recognize what to do?

Construct heat segments SW and SM.You now have what? Two right triangles, SWI and SIM. They have actually right angles, ∠SIW and also ∠SIM.Identify WI and IM as congruent, due to the fact that they room the two components of line segment WM the were bisected through SI.Identify SI as congruent to chin (by the reflexive property).

What walk that provide you? two congruent sides and also an included angle, which is what postulate? The SAS Postulate, of course! Therefore, heat segment SW ≅ SM.

So, did girlfriend sink or SWIM?

Converse the the Perpendicular Bisector Theorem

Notice that the theorem is created as one "if, then" statement. The immediately suggests you can write the converse that it, by switching the parts:


We can show this, too. Construct a heat segment HD. Ar a random point above it (but still somewhere in between Points H and D) and also call that Point T. If Point T is the exact same distance from Points H and also D, this converse statement says it have to lie ~ above the perpendicular bisector the HD.


You deserve to prove or disprove this by dropping a perpendicular line from Point T through line segment HD. Whereby your perpendicular line crosses HD, speak to it Point U.

If Point T is the same distance indigenous Points H and D, climate HU ≅ UD. If Point T is not the same distance native Points H and also D, then HU ≇ UD.

You deserve to go with the steps of developing two right triangles, △THU and △TUD and also proving angles and sides congruent (or no congruent), the same as with the initial theorem.

You would recognize the appropriate angles, the congruent sides along the original line segment HD, and the reflexive congruent next TU. As soon as you obtained to a pair of equivalent sides the were no congruent, then you would know Point T was no on the perpendicular bisector.

Only points lying on the perpendicular bisector will certainly be equidistant native the endpoints the the line segment. Everything else lands with a THUD.

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Lesson Summary

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