chapter 5 – special segments in triangles
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/1.jpg)
OBJECTIVE : 1) BE ABLE TO IDENTIFY THE MEDIAN AND
ALTITUDE OF A TRIANGLE2) BE ABLE TO APPLY THE MID-SEGMENT THEOREM3) BE ABLE TO USE TRIANGLE MEASUREMENTS TO
FIND THE LONGEST AND SHORTEST SIDE.
Chapter 5 – Special Segments in Triangles
![Page 2: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/2.jpg)
Figure Picture Definition IntersectionA 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.
![Page 3: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/3.jpg)
Figure Picture Definition IntersectionA 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
A C
2DE AC
![Page 4: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/4.jpg)
Example
1) Given: JK and KL are midsegments. Find JK and AB.
10
6
J
C
K
B
A L
5JK 12AB
![Page 5: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/5.jpg)
Example2) Find x.
73 x
73 x
67 x
2 3 7 7 6
6 14 78
6
x x
xx
x
3 7x
![Page 6: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/6.jpg)
Perpendicular Bisector Construction – pg. 264
1. 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.
![Page 7: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/7.jpg)
Given segment
perpendicular bisector
PA B
C
Thm 5.1:Perpendicular Bisector Thm
Thm 5.2: Converse of the Perpendicular Bisector ThmIf 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.
D is on CP
P
A B
C
D
![Page 8: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/8.jpg)
Theorem 5.5 Concurrency of Perpendicular Bisectors of a
TriangleThe perpendicular
bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.
BA = BD = BC
m DB = 3.09 cm
m CB = 3.09 cm
m AB = 3.09 cm
B
D
C
A
![Page 9: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/9.jpg)
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 mBAD = mCAD.
B
A
C
D
![Page 10: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/10.jpg)
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 = PFE
D
F P
B
A
C
![Page 11: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/11.jpg)
11
THEOREM 5.7 Concurrency of Medians of a TriangleThe 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
PE
D
F
B
A
C
![Page 12: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/12.jpg)
12
Example3) Find the
coordinates of the centroid of ∆JKL.
P
N
J (7, 10)
M
K (5, 2)
L (3, 6)
![Page 13: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/13.jpg)
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.
H
EA
C
BF
D
![Page 14: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/14.jpg)
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
ED
C
A B
![Page 15: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/15.jpg)
Example
4) Show that the midsegment MN is parallel to side JK and is half as long.
4
2
-2
-4
5 10
M
N
L (6, -1)
K (4, 5)
J (-2, 3)
1 2 1 2
Hint: Midpoint
( , ) ,2 2
x x y yM x y
![Page 16: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/16.jpg)
Theorems 5.10-5.11The longest side of a triangle is always opposite the largest angle and the smallest side is always opposite the smallest angle.
![Page 17: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/17.jpg)
Example
5) Write the measurements of the triangles from least to greatest.
H
J
G
45°
100°
35°
![Page 18: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/18.jpg)
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
1
C
A
B
![Page 19: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/19.jpg)
Example
6) Name the shortest and longest sides of the triangle below.
7) Name the smallest and largest angle of the triangle below.
![Page 20: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/20.jpg)
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
![Page 21: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/21.jpg)
Example9) If two sides of a triangle measure 5 and 7, what are the possible measures for the third side?
12 2x
![Page 22: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/22.jpg)
READ 264-267 , 272-274 , 279-281 , 287-289 , 295-297
DEFINE: MEDIAN, ALTITUDE, PERPENDICULAR BISECTOR, ANGLE
BISECTOR, MIDSEGMENT, CIRCUMCENTER, INCENTER, ORTHOCENTER, CENTROID
ASSIGNMENT
![Page 23: Chapter 5 – Special Segments in Triangles](https://reader033.vdocuments.mx/reader033/viewer/2022061615/568163ee550346895dd5659d/html5/thumbnails/23.jpg)
Class Activity
Page 269 #21-26Page 276 #14-17Page 282 #8-12, 17-20Page 290 #12-17Page 298 #6-11