Perpendicular Bisectors and Altitudes of Triangles

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Presentation transcript:

Perpendicular Bisectors and Altitudes of Triangles

Perpendicular Bisector Perpendicular Bisector- A segment, line, ray, or plane that is __________ to another segment at it’s __________. Equidistant- A point is equidistant from two figures if the point is the same _________ from each figure.

Perpendicular Bisector Theorem In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. C Ex. If CD AB And Then A D B T

Converse of Perpendicular Bisector Theorem In a plane, if a point is ________ from the endpoints of a segment then it is on the __________________ of the segment. C A B

Perpendicular Bisectors of a Triangle A line, ray, or segment that is perpendicular to a side of the triangle at the midpoint of the side.

Point of Concurrency Circumcenter- the point of concurrency for the 3 perpendicular bisectors of a triangle. Right Acute Obtuse Circumcenter Circumcenter Circumcenter lies on lies inside lies outside hypotenuse triangle triangle

Circumcenter It is the same distance from each vertex of the triangle to the circumcenter. Ex. A G B C

Example 1 NO is the perpendicular bisector of LM. If OM = 4 and LN = 6, find LO and MN. L N M O

Example 2 2. NO is the perpendicular bisector of LM. If MN = 6x + 18 and LN = 8x + 6, find LN and MN. L N M O

Example 3 3. The perpendicular bisectors of LMN meet at K. Find LK L N 7 4.2

Altitudes of Triangles Altitude- the perpendicular segment from a vertex of the triangle to the opposite side or to a line that contains the opposite side of a triangle. A AD is an altitude of triangle ABC. B D C

Point of Concurrency Orthocenter- the point of concurrency for the 3 altitudes of a triangle. Right Acute Obtuse Orthocenter Orthocenter Orthocenter lies on lies inside lies outside hypotenuse triangle triangle

There is absolutely nothing special about the Orthocenter!!!

Check What are the two differences between the perpendicular bisector and the altitude?

Example 4 Is BD a perpendicular bisector of ABC? Is BD an altitude of ABC? B D A C B A D C B A C D

Example 5 Find the orthocenter of the triangle. (3, 7) (8, 3) (0, 0)