4.5 Special Segments in Triangles
- 1. Special Segments in Triangles The student is able to (I can): • Construct perpendicular and angle bisectors • Use bisectors to solve problems • Identify the circumcenter and incenter of a triangle • Identify altitudes and medians of triangles • Identify the orthocenter and centroid of a triangle • Use triangle segments to solve problems
- 2. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. P D A E PD = AD PE = AE
- 3. Converse of Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. S K Y T ST = YT KT SY ⊥
- 4. Examples Find each measure: 1. YO 2. GR B O Y 15 G I R L 20 20 2x-1 x+8
- 5. Examples Find each measure: 1. YO YO = BO = 15 2. GR B O Y 15 G I R L 20 20 2x-1 x+8 2x – 1 = x + 8 x = 9 GR = 2x – 1 + x + 8 = 34
- 6. Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem If a point is equidistant from the sides of an angle, then it is on the angle bisector. A L G N AN = GN ALN GLN
- 7. circumcenter – the intersection of the perpendicular bisectors of a triangle.
- 8. circumcenter – the intersection of the perpendicular bisectors of a triangle. It is called the circumcenter, because it is the center of a circle that circumscribes the triangle (all three vertices are on the circle).
- 9. incenter – the intersection of the angle bisectors of a triangle.
- 10. incenter – the intersection of the angle bisectors of a triangle. It is called the incenter because it is the center of the circle that is inscribed in the circle (the circle just touches all three sides).
- 11. median – a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. altitude – a perpendicular segment from a vertex to the line containing the opposite side.
- 12. centroid – the intersection of the medians of a triangle. It is also the center of mass for the triangle.
- 13. Centroid Theorem The centroid of a triangle is located of the distance from each vertex to the midpoint of the opposite side. G H J X Y Z R 2 3 GR GY = 2 3 HR HZ = 2 3 JR JX = 2 3
- 14. orthocenter – the intersection of the altitudes of a triangle.