Chapter 5 – Special Segments in Triangles

1. Chapter 5 – Special Segments in Triangles Objective: 1) Be able to identify the median and altitude of a triangle 2) Be able to apply the Mid-segment Theorem 3) Be able to use triangle measurements to find the longest and shortest side.

2. Median Altitude Perpendicular Bisector

3. Angle Bisector B Midsegment E D A C

4. Example 1) Given: JK and KL are midsegments. Find JK and AB. B K J 6 L A C 10

5. Example 2) Find x.

6. Perpendicular Bisector Construction – pg. 264 • Draw a line m. Label a point P in the middle of the line. • Place compass point at P. Draw an arc that intersects line m twice. Label the intersections as A and B. • Use a compass setting greater than AP. Draw an arc from A. With the same setting, draw an arc from B. Label the intersection of the arcs as C. • Use a straightedge to draw CP. This line is perpendicular to line m and passes through P.

7. Thm 5.1:Perpendicular Bisector Thm Thm 5.2: Converse of the Perpendicular Bisector Thm If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If DA = DB, then D lies on the perpendicular bisector of AB. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the perpendicular bisector of AB, then CA = CB.

8. The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. BA = BD = BC Theorem 5.5 Concurrency of Perpendicular Bisectors of a Triangle

9. If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. If DB = DC, then mBAD = mCAD. Theorem 5.3 Angle Bisector Theorem

10. The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. PD = PE = PF Theorem 5.6 Concurrency of Angle Bisectors of a Triangle

11. The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. If P is the centroid of ∆ABC, then AP = 2/3 AD, BP = 2/3 BF, and CP = 2/3 CE THEOREM 5.7 Concurrency of Medians of a Triangle

12. 3) Find the coordinates of the centroid of ∆JKL. Example

13. The lines containing the altitudes of a triangle are concurrent. If AE, BF, and CD are altitudes of ∆ABC, then the lines AE, BF, and CD intersect at some point H. Theorem 5.8 Concurrency of Altitudes of a Triangle

14. The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. DE ║ AB, and DE = ½ AB Theorem 5.9: Midsegment Theorem

15. 4) Show that the midsegment MN is parallel to side JK and is half as long. Example

16. Theorems 5.10-5.11 • The longest side of a triangle is always opposite the largest angle and the smallest side is always opposite the smallest angle.

17. 5) Write the measurements of the triangles from least to greatest. Example 100° 45° 35°

18. Theorem 5.12-Exterior Angle Inequality • The measure of an exterior angle of a triangle is greater than the measure of either of the two non- adjacent interior angles. • m1 > mA and m1 > mB

19. Example 6) Name the shortest and longest sides of the triangle below. 7) Name the smallest and largest angle of the triangle below.

20. Theorem 5.13 - Triangle Inequality Thm. • The sum of the lengths of any two sides of a triangle is greater than then length of the third side. Example: 8) Determine whether the following measurements can form a triangle. • 8, 7, 12 • 2, 5, 1 • 9, 12, 15 • 6, 4, 2 YES NO YES NO

21. Example 9) If two sides of a triangle measure 5 and 7, what are the possible measures for the third side?

22. ASSIGNMENT Read 264-267, 272-274, 279-281, 287-289, 295-297 Define: Median, Altitude, Perpendicular Bisector, Angle Bisector, Midsegment, Circumcenter, Incenter, Orthocenter, Centroid

23. Class Activity Page 269 #21-26 Page 276 #14-17 Page 282 #8-12, 17-20 Page 290 #12-17 Page 298 #6-11