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. Chapter 5 – Special Segments in Triangles

FigurePictureDefinitionIntersection A segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. The concurrence of the medians is called the centroid. The perpendicular segment from a vertex to the opposite side. The concurrence of the altitudes is called the orthocenter. A segment, line or ray that is perpendicular to a side and passes through the midpoint. The concurrence of the perpendicular bisectors is called the circumcenter.

FigurePictureDefinitionIntersection A ray that divides an angle into two adjacent angles that are congruent. The concurrence of the angle bisectors is called the incenter. A segment that connects the midpoints of two sides of a triangle. The midsegment of a triangle is parallel to the side it does not touch and is half as long. B D E AC

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

Example 2) Find x.

Perpendicular Bisector Construction – pg Draw a line m. Label a point P in the middle of the line. 2. Place compass point at P. Draw an arc that intersects line m twice. Label the intersections as A and B. 3. 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. 4. Use a straightedge to draw CP. This line is perpendicular to line m and passes through P.

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.

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

Theorem 5.3 Angle Bisector Theorem 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 5.6 Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. PD = PE = PF

11 THEOREM 5.7 Concurrency of Medians of a Triangle 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

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

13 Theorem 5.8 Concurrency of Altitudes of a Triangle 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 5.9: Midsegment Theorem  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

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

Theorems The longest side of a triangle is always opposite the largest angle and the smallest side is always opposite the smallest angle.

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

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

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

Theorem 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

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

READ , , , , DEFINE: MEDIAN, ALTITUDE, PERPENDICULAR BISECTOR, ANGLE BISECTOR, MIDSEGMENT, CIRCUMCENTER, INCENTER, ORTHOCENTER, CENTROID ASSIGNMENT

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