Perpendicular Bisectors and Altitudes of Triangles Perpendicular Bisector
Perpendicular Bisectors and Altitudes of Triangles
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 Circumcenter lies on hypotenuse Acute Circumcenter lies inside triangle Obtuse Circumcenter lies outside triangle
Circumcenter It is the same distance from each vertex of the triangle to the circumcenter. Ex. A G B C
Example 1 1. NO is the perpendicular bisector of LM. If OM = 4 and LN = 6, find LO and MN. N L O M
Example 2 2. NO is the perpendicular bisector of LM. If MN = 6 x + 18 and LN = 8 x + 6, find LN and MN. N L O M
Example 3 3. The perpendicular bisectors of meet at K. Find LK N 7 K L 4. 2 M LMN
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 Orthocenter lies on hypotenuse Acute Obtuse Orthocenter lies inside triangle Orthocenter lies outside triangle
Orthocenter 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 Is BD an altitude of ABC? B B A A D C D B C A D C
Example 5 Find the orthocenter of the triangle. (3, 7) (0, 0) (8, 3)
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