5.3: Concurrent Lines, Medians and Altitudes Objectives: To identify properties of perpendicular bisectors and angle bisectors To identify properties of medians and altitudes of triangles

Vocabulary Concurrent: 3 or more lines intersect at a point Point of Concurrency: where the lines intersect

Circumcenter: The point of concurrency of the perpendicular bisectors of a triangle  If you were to draw a circle around a triangle, where each vertex of the triangle are points on the circle, the circle would be circumscribed about that triangle  The circumcenter of the triangle is ALSO the center of the circle circumscribed about it

Perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices of the triangle

Perpendicular Bisectors Acute triangle: Circumcenter inside triangle Right triangle: Circumcenter lies ON the triangle Obtuse Triangle: Circumcenter is outside triangle

Find the center of the circle that you can circumscribe about the triangle with vertices (0, 0), (-8,0) and (0,6). 1.Plot the points on a coordinate plane. 2.Draw the triangle 3.Draw the perpendicular bisectors of at least 2 sides. 4.The circumcenter of this triangle will be the center of the circle.

Incenter of a triangle: point of concurrency of the angle bisectors A circle is inscribed in a triangle when the circle is tangent to each side (also called an incircle) The incenter of a triangle is the center of an inscribed circle

ANGLE BISECTORS of a triangle are concurrent at a point equidistant to the sides of the triangle. (The incenter is equidistant to the sides)  The incenter of ALL types of triangles is always INSIDE the triangle.

Example: Find the values of the variables. y 4 x 10

Median of a Triangle A segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side

WHAT IS THE DIFFERENCE BETWEEN A MEDIAN AND A PERPENDICULAR BISECTOR??

CENTROID: Point of concurrency of medians of a triangle The medians of a triangle intersect at a point that is 2/3 the distance from the vertex to the midpoint of the opposite side. G is the centroid AG = AD BG = BF CG= CE

O is the centroid of triangle ABC. CE = 11 FO = 10 CO = 8 1.Find EB. 2.Find OD. 3.Find FB. 4.Find CD. 5.If AO = 6, find OE.

Altitude of a Triangle The perpendicular segment from the vertex to the line containing the opposite side Acute Triangle: Inside Right Triangle: Side Obtuse Triangle: Outside

What is the difference between an altitude and a perpendicular bisector?

Orthocenter of a Triangle Point of concurrency for altitudes of a triangle Acute Triangle: Inside Right Triangle: Vertex Obtuse Triangle: Outside