Perpendicular Bisector Theorem (Proof, Converse, & Examples)
All good learning begins via vocabulary, so we will certainly focus on the 2 necessary words of the theorem. Perpendicular indicates two line segments, rays, lines or any type of combicountry of those that meet at ideal angles. A line is perpendicular if it intersects another line and also creates right angles.
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A bisector is a things (a line, a ray, or line segment) that cuts one more object (an angle, a line segment) into 2 equal components. A bisector cannot bisect a line, bereason by interpretation a line is infinite.
Putting the 2 interpretations together, we gain the concept of a perpendicular bisector, a line, ray or line segment that bisects an angle or line segment at a ideal angle.
Before you get all bothered around it being a perpendicular bisector of an angle, consider: what is the meacertain of a directly angle? 180°; that means a line dividing that angle right into 2 equal components and also forming 2 right angles is a perpendicular bisector of the angle.
Perpendicular Bisector Theorem
Okay, we lhelp the groundwork-related. So putting everything together, what does the Perpendicular Bisector Theorem say?
How Does It Work?
Suppose you have actually a large, square plot of land also, 1,000 meters on a side. You developed a humdinger of a radio tower, 300 meters high, right smack in the middle of your land also. You plan to broadactors rock music day and night.
Anymeans, that location for your radio tower means you have actually 500 meters of land to the left, and also 500 meters of land to the right. Your radio tower is a perpendicular bisector of the length of your land.
You have to reinforce the tower through wires to keep it from tipping over in high winds. Those are dubbed man wires. How long have to a male wire from the peak down to the land be, on each side?
Due to the fact that you built a perpendicular bisector, you carry out not have to measure on each side. One measurement, which you have the right to calculate making use of geometry, is sufficient. 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 finish. The tower meets your land also at 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 whopping 583.095 meters long to run from the peak of the tower to the edge of your land also. You repeat the operation at the 200 meter elevation, and also the 100 meter elevation.
For eextremely elevation you pick, you will certainly cut man wires of the same lengths for the left and best side of your radio tower, bereason 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 via just a line segment, HM, and also its perpendicular bisector, WA, we have the right to prove this theorem.
We are offered line segment HM and also we have bisected it (separated it exactly in two) by a line WA. That line bisected HM at 90° because it is a given. This indicates, if we run a line segment from Point W to Point H, we deserve to develop best triangle WHA, and also an additional line segment WM creates right triangle WAM.
What carry out we have now? We have actually 2 ideal triangles, WHA and WAM, sharing side WA, with all these congruences:WA ≅ WA (by the reflexive property)∠WAH ≅ ∠WAM (90° angles; given)HA ≅ AM (bisector; given)
What does that look like? We hope you sassist Side Angle Side, because that is exactly what it is.
That suggests sides WH and WM are congruent, because CPCTC (matching parts of congruent triangles are congruent). WHAM! Proven!
You can tackle the theorem yourself now. You will certainly either sink or swim on this one. Here is a line segment, WM. We construct a perpendicular bisector, SI.
How can you prove that SW ≅ SM? Do you recognize what to do?Construct line segments SW and also SM.You currently have what? Two best triangles, SWI and also SIM. They have actually ideal angles, ∠SIW and ∠SIM.Identify WI and also IM as congruent, bereason they are the 2 parts of line segment WM that were bisected by SI.Identify SI as congruent to itself (by the reflexive property).
What does that give you? Two congruent sides and an consisted of angle, which is what postulate? The SAS Postulate, of course! As such, line segment SW ≅ SM.
So, did you sink or SWIM?
Converse of the Perpendicular Bisector Theorem
Notice that the theorem is created as an "if, then" statement. That instantly says you deserve to write the converse of it, by switching the parts:
We have the right to present this, also. Construct a line segment HD. Place a random point over it (yet still somewbelow between Points H and also D) and also call it Point T. If Point T is the same distance from Points H and also D, this converse statement claims it should lie on the perpendicular bisector of HD.
You can prove or disprove this by dropping a perpendicular line from Point T with line segment HD. Wbelow your perpendicular line crosses HD, contact it Point U.
If Point T is the very same distance from Points H and D, then HU ≅ UD. If Point T is not the same distance from Points H and D, then HU ≇ UD.
You can go with the actions of creating 2 right triangles, △THU and △TUD and proving angles and also sides congruent (or not congruent), the very same as through the original theorem.
You would recognize the ideal angles, the congruent sides along the original line segment HD, and the reflexive congruent side TU. When you obtained to a pair of corresponding sides that were not congruent, then you would certainly recognize Point T was not on the perpendicular bisector.
Only points lying on the perpendicular bisector will be equifar-off from the endpoints of the line segment. Everypoint else lands with a THUD.
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After you functioned your method via all the angles, proofs and multimedia, you are now able to respeak to the Perpendicular Bisector Theorem and also test the converse of the Theorem. You likewise gained a refresher in what "perpendicular," "bisector," and also "converse" expect.