Chapter 5: Relationships in Triangles. Lesson 5.1 Bisectors, Medians, and Altitudes.

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5.1 Bisector, Medians, and Altitudes

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Presentation transcript:

Chapter 5: Relationships in Triangles

Lesson 5.1 Bisectors, Medians, and Altitudes

Perpendicular Bisector DefinitionFacts to KnowPoint of Concurrency Example A line, segment or ray that passes through the midpoint of the opposite side and is perpendicular to that side Any point on a perpendicular bisector is equidistant from the endpoints Circumcenter: The point where 3 perpendicular bisectors intersect - the circumcenter is equidistant from all vertices of the triangle DC B A BD = CD AD BC E is the circumcenter- AE = BE = CE E

Median DefinitionFacts to KnowPoint of Concurrency Example A segment that goes from a vertex of the triangle to the midpoint of the opposite side The median splits the opposite side into two congruent segments Small = 1/3 median Big = 2/3 median 2 x small = big Centroid: The point where 3 medians intersect DC B A BD = CD E is the centroid- ED = 1/3 AD AE = 2/3 AD 2 ED = AE E

Angle Bisector DefinitionFacts to KnowPoint of Concurrency Example A line, segment, or ray that passes through the middle of an angle and extends to the opposite side Any point on an angle bisector is equidistant from the sides of the triangle Incenter: The point where 3 angle bisectors intersect -the incenter is equidistant from all sides of the triangle DC B A FE BAD = CAD G is the incenter- EG = FG G

Altitude DefinitionFacts to KnowPoint of Concurrency Example A segment that goes from a vertex of the triangle to the opposite side and is perpendicular to that side Orthocenter: The point where 3 altitudes intersect DC B A AD BC

C. Find the measure of EH.

A. Find QS. B. Find  WYZ.

In the figure, A is the circumcenter of ΔLMN. Find x if m  APM = 7x + 13.

In the figure, point D is the incenter of ΔABC. What segment is congruent to DG?

In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.