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

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

Vocabulary Perpendicular Bisectors Angle Bisectors Medians Altitudes Points of Concurrency

Perpendicular bisector 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

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.

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

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.

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.

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.

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.

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.

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.

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.

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 Segments Point of Concurrency Angle Bisectors Incenter Perpendicular Bisectors Circumcenter Altitude Orthocenter Median Centroid

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.

Facts To Remember & MEMORIZE! 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

Points of Concurrency (con’t)

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