Geo: Sec 5.1 Perpendiculars & Bisectors Thm 5.1 Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant

Geo: Sec 5.1 Perpendiculars & Bisectors

Thm 5.1 Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.



equidistant = its distance from each endpoint is the same

perpendicular bisector = ray, segment, line, or plane that is perpendicular to the segment at its midpoint



If CP is the perpendicular bisector of AB, then CA = CB.

Sec 5.1 Perpendiculars & Bisectors

Thm 5.2 Converse of Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.


Thm 5.2 Converse of Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.

If DA = DB, then D lies on the perpendicular bisector of AB.


In the diagram shown, MN is the perpendicular bisector of ST.

What segments lengths in thediagram are equal?



What segments lengths in thediagram are equal?

Solution: MN bisects ST, so NS = NT. Because M is on the perpendicular bisector of ST, MS = MT. (Thm 5.1)



Explain why Q is on MN.



Explain why Q is on MN.

Solution: QT = QS so Q is equidistant from S and T. By Thm 5.2, Q is on the perpendicular bisector of ST, which is MN.


Thm 5.3 Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.


Thm 5.3 Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

If m < BAD = m < CAD, then DB = DC.


Thm 5.4 Converse of Angle Bisector Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.


Thm 5.4 Converse of Angle Bisector Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

If DB = DC, thenm < BAD = m < CAD,


Roof Trusses: Some roofs are built with wooden trusses that are assembled in the factory and shipped to the building site. In the diagram of the roof truss shown below, you are given that AB bisects <CAD and that < ACB and <ADB are right angles.



What can you say about BC and BD ?



What can you say about BC and BD ?Solution: Because BC and BD meetAC and AD @ right angles, they are perpendicular segments to the sides of <CAD. This implies that their lengths represent the distances from pt. B to AC and AD.



What can you say about BC and BD ?Solution: Because BC and BD meetAC and AD @ right angles, they are perpendicular segments to the sides of <CAD. This implies that their lengths represent the distances from pt. B to AC and AD. Because pt B is on the bisector of <CAD, it is equidistant from the sides of the angle.



Solution: Because BC and BD meetAC and AD @ right angles, they are perpendicular segments to the sides of <CAD. This implies that their lengths represent the distances from pt. B to AC and AD. Because pt B is on the bisector of <CAD, it is equidistant from the sides of the angle. So BC = BDand BC = BD.~

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Geo: Sec 5.1 Perpendiculars & Bisectors Thm 5.1 Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant