Perpendicular Bisectors of a Triangle Defn: Perpendicular Bisector of a Triangle: A segment is a perpendicular bisector of a triangle iff it is the perpendicular bisector of a side of the triangle. 4/21/2017 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle Every triangle has 3 perpendicular bisectors. 4/21/2017 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle 4/21/2017 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle 4/21/2017 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles The 3 perpendicular bisectors of any triangle will intersect at a point that is equidistant from the vertices of the triangle. This point is called the circumcenter and is the center of a circle that contains all 3 vertices of the triangle. 4/21/2017 5-1: Special Segments in Triangles

Angle Bisectors of Triangles Defn: Angle Bisector of a Triangle: A segment is an angle bisector of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is any other point on the triangle such that the segment bisects an angle of the triangle. 4/21/2017 5-1: Special Segments in Triangles

Angle Bisectors of Triangles Every triangle has 3 angle bisectors which will always intersect in the same point - the incenter. The incenter is the same distance from all 3 sides of the triangle. The incenter of a triangle is also the center of a circle that will intersect each side of the triangle in exactly one point. 4/21/2017 5-1: Special Segments in Triangles

Angle Bisectors of Triangles 4/21/2017 5-1: Special Segments in Triangles

Angle Bisectors of Triangles 4/21/2017 5-1: Special Segments in Triangles

Angle Bisectors of Triangles 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles RU is an angle bisector, m∠RTU = 13x – 24, m∠TRS = 12x – 34 and m∠RUS = 92. Determine m∠RSU. Is RU ⊥ TS? R S T U 4/21/2017 5-1: Special Segments in Triangles

Angle Bisector Theorem If D is on the bisector of ∠ABC, then X D B Y C DX = DY. 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles Median of a Triangle Defn: Median of a Triangle: A segment is a median of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is the midpoint of the side opposite that vertex. 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles Medians of a Triangle Every triangle has 3 medians. 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles Medians of a Triangle Every triangle has 3 medians. 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles Medians of a Triangle Every triangle has 3 medians. 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles Centroids The medians of a triangle will always intersect at the same point - the centroid. The centroid of a triangle is located 2/3 of the distance from the vertex to the midpoint of the opposite side. 4/21/2017 5-1: Special Segments in Triangles

Average of the three vertices Formula for finding the centroid of a triangle   Average of the three vertices

(7, 10) (3, 6) (5, 2) Find the coordinates of the centroid of  JKL. Finding the Centroid of a Triangle Find the coordinates of the centroid of  JKL. SOLUTION J K L N P (7, 10) (3, 6) (5, 2)  

5-1: Special Segments in Triangles Centroid centroid 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles Centroid 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles Points U, V, and W are the midpoints of YZ, XZ and XY respectively. Find a, b, and c. 4/21/2017 5-1: Special Segments in Triangles

Altitudes of Triangles Defn: Altitude of a Triangle: A segment is an altitude of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is on the line containing the opposite side such that the segment is perpendicular to line. 4/21/2017 5-1: Special Segments in Triangles

Altitudes of Triangles Every triangle has 3 altitudes that will always intersect in the same point. 4/21/2017 5-1: Special Segments in Triangles

Altitudes of Triangles If the triangle is acute, then the altitudes are all in the interior of the triangle. 4/21/2017 5-1: Special Segments in Triangles

Altitudes of Triangles If the triangle is a right triangle, then one altitude is in the interior and the other 2 altitudes are the legs of the triangle. 4/21/2017 5-1: Special Segments in Triangles

Altitudes of Triangles If the triangle is an obtuse triangle, then one altitude is in the interior and the other 2 altitudes are in the exterior of the triangle. 4/21/2017 5-1: Special Segments in Triangles

Altitudes of Triangles 4/21/2017 5-1: Special Segments in Triangles

ZC is an altitude, m∠CYW = 9x + 38 and m∠WZC = 17x. Find m∠WZC. 4/21/2017 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles In ΔABC below, AB ≅ BC and AD bisects ∠BAC. If the length of BD is 3(x + 2) units and BC = 42 units, what is the value of x? 5 6 12 13 A C D B 4/21/2017 5-1: Special Segments in Triangles