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.

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Presentation transcript:

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.

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

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. 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.

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

Sec 5.1 Perpendiculars & Bisectors In the diagram shown, MN is the perpendicular bisector of ST. What segments lengths in the diagram are equal?

Sec 5.1 Perpendiculars & Bisectors In the diagram shown, MN is the perpendicular bisector of ST. What segments lengths in the diagram are equal? Solution: MN bisects ST, so NS = NT. Because M is on the perpendicular bisector of ST, MS = MT. (Thm 5.1)

Sec 5.1 Perpendiculars & Bisectors In the diagram shown, MN is the perpendicular bisector of ST. Explain why Q is on MN.

Sec 5.1 Perpendiculars & Bisectors In the diagram shown, MN is the perpendicular bisector of ST. 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.

Sec 5.1 Perpendiculars & Bisectors 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.

Sec 5.1 Perpendiculars & Bisectors 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.

Sec 5.1 Perpendiculars & Bisectors 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.

Sec 5.1 Perpendiculars & Bisectors 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, then m < BAD = m < CAD,

Sec 5.1 Perpendiculars & Bisectors 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.

Sec 5.1 Perpendiculars & Bisectors 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 ?

Sec 5.1 Perpendiculars & Bisectors 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 ? Solution: Because BC and BD meet AC and 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.

Sec 5.1 Perpendiculars & Bisectors 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 ? Solution: Because BC and BD meet AC and 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.

Sec 5.1 Perpendiculars & Bisectors 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. Solution: Because BC and BD meet AC and 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 = BD and BC = BD. ~