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

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.

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

Example

1) Given: JK and KL are midsegments. Find JK and AB.

10

6

J

C

K

B

A L

5JK 12AB

Example2) Find x.

73 x

73 x

67 x

2 3 7 7 6

6 14 7

8

6

x x

x

x

x

3 7x

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.

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

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

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

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

FP

B

A

C

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

PE

D

F

B

A

C

12

Example

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

P

N

J (7, 10)

M

K (5, 2)

L (3, 6)

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

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

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

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.

Example

5) Write the measurements of the triangles from least to greatest.

H

J

G

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.

m1 > mA and m1 > mB

1

C

A

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 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?

12 2x

READ 264-267 , 272-274 , 279-281 , 287-289 , 295-297

DEFINE: MEDIAN, ALTITUDE, PERPENDICULAR BISECTOR, ANGLE

BISECTOR, MIDSEGMENT, CIRCUMCENTER, INCENTER, ORTHOCENTER, CENTROID

ASSIGNMENT

Class Activity

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