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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.
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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
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 SegmentsPoint of Concurrency Perpendicular Bisectors Circumcenter Angle Bisectors Incenter Median Centroid Altitude Orthocenter
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.
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!
The End (Finally!) Study Chapter 6!!!!!!