1. 2.1b: Triangle Properties -Special Segments in Triangles M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts GSE’s G-CO.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. CCSS

2. Median Connects a vertex to the of the opposite side MIDDLE A B C Tells us that it cut the side BC in half so BD DC D E F When you combine ALL the medians of one triangle, you get the CENTROID A centroid would balance the triangle if you held it up with a pencil

3. Example Determine the coordinates of J so that SJ is a median of the triangle. J ( 9, 0.5) Ans: Use the midpoint formula for GB J =

4. Connects the vertex to where it is perpendicular to the opposite side Altitude W A P R T Z Combine all three ALTIUDE’S in a triangle and you get a ORTHOCENTER Tells us the segment is perpendicular

5. Altitude with an obtuse triangle

6. Example BD is an altitude of Triangle ABC Find BC, and AC Ans: 7x+20=90 7x = 70 X = 10 But wait….. Re-read the question Find BC, so BC = 3(10)-5 = 25 3x-5

7. Angle Bisector A segment that cuts one vertex angle in half and goes to the opposite side of the triangle A B C Draw angle bisector AF F Indicates the angle A was bisected

8. Example W B N Find m NWB if WT is an angle bisector Of WNB m NWT = 3x + 8 m NWB = 3x + 34 T ANS: Draw your diagram 3x+8 3x+8 + 3x+8 = 3x + 34 6x + 16 = 3x + 34 3x = 18 x = 6 3(6) + 34 = 52

9. Perpendicular Bisector A segment that: 1) is perpendicular to one side of the triangle 2) bisects the same side H BO Draw the segment AT that is a perpendicular bisector of BO on Triangle HBO

10. example Name all segments that are (if any) Angle Bisectors Perpendicular Bisectors Altitudes Medians QU NONE RT SP

11. Example has vertices A(-4,1), B ( 1, 6) and C (3, -4). Find: 1) The coordinate of T if it is on AB and TR is a perpendicular bisector to side AB 2) Find the slope of TR

12. Activity

18. End of notes