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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. Median. Altitude. Perpendicular Bisector. Angle
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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.
Median Altitude Perpendicular Bisector
Angle Bisector B Midsegment E D A C
Example 1) Given: JK and KL are midsegments. Find JK and AB. B K J 6 L A C 10
Example 2) Find x.
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.
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.
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
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 mBAD = mCAD. Theorem 5.3 Angle Bisector Theorem
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
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
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
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
4) Show that the midsegment MN is parallel to side JK and is half as long. Example
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.
5) Write the measurements of the triangles from least to greatest. Example 100° 45° 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. • m1 > mA and m1 > mB
Example 6) Name the shortest and longest sides of the triangle below. 7) Name the smallest and largest angle of the triangle below.
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
Example 9) If two sides of a triangle measure 5 and 7, what are the possible measures for the third side?
ASSIGNMENT Read 264-267, 272-274, 279-281, 287-289, 295-297 Define: Median, Altitude, Perpendicular Bisector, Angle Bisector, Midsegment, Circumcenter, Incenter, Orthocenter, Centroid
Class Activity Page 269 #21-26 Page 276 #14-17 Page 282 #8-12, 17-20 Page 290 #12-17 Page 298 #6-11