Relationships in Triangles

1. Relationships in Triangles Bisectors, Medians, and Altitudes Section 6.1 – 6.3 Students Should Begin Taking Notes At Screen 4!!

2. Objectives of this lesson • To identify and use perpendicular bisectors & angle bisectors in triangles • To identify and use medians & altitudes in triangles

3. Vocabulary • Perpendicular Bisectors • Angle Bisectors • Medians • Altitudes • Points of Concurrency

4. STUDENTS SHOULD BEGIN TAKING NOTES HERE! Perpendicular bisector • A line segment or a ray that passes through the midpoint of a side of a triangle & is ⊥ to that side. In the picture to the right, the red line segment is the ⊥ bisector

5. Perpendicular Bisector (con’t) • For every triangle there are 3 perpendicular bisectors • The 3 perpendicular bisectors intersect in a common point named the circumcenter. In the picture to the right point K is the circumcenter.

6. Perpendicular Bisector (con’t) • Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment • Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment

7. Angle Bisector • A line, line segment or ray that bisects an interior angle of a triangle In the picture to the right, the red line segment is the angle bisector. The  arc marks show the 2  angles that were formed when the angle bisector bisected the original angle.

8. Angle Bisector (con’t) • For every triangle there are 3 angle bisectors. • The 3 angle bisectors intersect in a common point named the incenter In the picture to the right, point I is the incenter.

9. Angle Bisector (con’t) • Any point on the angle bisector is equidistant from the sides of the angle. • Any point equidistant from the sides of an angle lies on the angle bisector.

10. Median A line segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex. In the picture to the right, the blue line segment is the median.

11. Median (con’t) • For every triangle there are 3 medians • The 3 medians intersect in a common point named the centroid In the picture to the right, point O is the centroid.

12. Altitudes A line segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. In the picture above, ∆ABC is an obtuse triangle & ∠ACB is the obtuse angle. BH is an altitude.

13. Altitudes (con’t) • For every triangle there are 3 altitudes • The 3 altitudes intersect in a common point called the orthocenter. In the picture to the right, point H is the orthocenter.

14. Points of Concurrency Concurrent Lines 3 or more lines that intersect at a common point Point of Concurrency The point of intersection when 3 or more lines intersect. Type of Line SegmentsPoint of Concurrency Angle Bisectors Incenter Perpendicular Bisectors Circumcenter Altitude Orthocenter Median Centroid

15. Points of Concurrency (con’t) Facts to remember: • The circumcenter of a triangle is equidistant from the vertices of the triangle. • Any point on the angle bisector is equidistant from the sides of the angle (Converse of #3) • Any point equidistant from the sides of an angle lies on the angle bisector. (Converse of #2) • The incenter of a triangle is equidistant from each side of the triangle. • The distance from a vertex of a triangle to the centroid is 2/3 of the median’s entire length. The length from the centroid to the midpoint is 1/3 of the length of the median.

16. 1. Perpendicular Bisectors 2. Angle Bisectors 3. Medians 4. Altitudes 1. …form right angles AND 2  lines segments 2. …form 2  angles 3. …form 2  line segments 4. … form right angles Facts To Remember & MEMORIZE!

17. Points of Concurrency (con’t)

18. The End (Finally!) Study Chapter 6!!!!!!