1. MEDIANS, ALTITUDES, AND PERPENDICULAR BISECTORS October 13, 2009

2. Objectives  Content  Students will learn about medians, altitudes, perpendicular bisectors, and angle bisectors.  Language  Students will participate in discussion.  Students will demonstrate understanding in writing.

3. Yesterday’s Homework

12. A little vocabulary review  Scalene triangle:________sides are congruent  Isosceles triangle:________sides are congruent  Equilateral triangle:________sides are congruent

13. More vocabulary review  Acute: ______ acute angle(s)  Obtuse: ______ obtuse angle(s)  Right: ______ right angle(s)  Equiangular: ______ angles are congruent

14. Find these measures.  Find the missing values. x xx x 35

15. And these…  Find the missing values.

16. What does it mean to be congruent?  When two figures have the same shape and size, they are called congruent.  What kinds of congruent triangles are there?

17. What are the parts of an isosceles triangle?  Base  Legs  Vertex angle  Base angles

18. What is a median?  A median of a triangle is a segment from a vertex to the midpoint of the opposite side.

19. What is an altitude?  An altitude of a triangle is the perpendicular segment from a vertex to a line that contains the opposite side.

20. More about altitudes  In a right triangle, two of the altitudes are part of the triangle.  Since the two legs are perpendicular to each other then each leg is also an altitude of the triangle.  The third altitude is inside the triangle.

21. Even more about altitudes  In an obtuse triangle, two of the altitudes are actually outside the triangle.  This is why the definition states that the altitude is the perpendicular segment from a vertex to the line that contains the opposite side.

22. What is a perpendicular bisector?  A perpendicular bisector is a line (or ray or segment) that is perpendicular to a segment at its midpoint.

23. What is an angle bisector?  An angle bisector cuts an angle into two equal parts.

24. What are concurrent line?  Concurrent lines are two or more lines that intersect in one point.

25. Concurrent medians  The medians are concurrent at a point called the centroid. The length of the segment from a vertex to the centroid is always 2/3 the length of the entire median.

26. Concurrent altitudes  The lines that contain the altitudes are concurrent at a point called the orthocenter.

27. Concurrent perpendicular bisectors  The perpendicular bisectors of the sides are concurrent at a point called the circumcenter.

28. Concurrent angle bisectors  Angle bisectors are rays running from the bisector of each angle of the triangle. They all meet at the incenter.

29. Working with these terms on a coordinate plane  Midpoint formula =

30. Try this  Triangle RST has the following coordinates: R (-6, -8), S (6, 4), T (-6, 10)  Find the midpoint of each side.  What else can we find?