5.1 MIDSEGMENTS OF TRIANGLES
THEOREM 5-1: TRIANGLE MIDSEGMENT THEOREM
Theorem If Then
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half the distance.
D is the midpoint of CA andE is the midpoint of CB
1
2
DE AB and
DE AB
A B
C
D E
//
A B
C
D E
2x
x
EXAMPLE 1 Find all missing side lengths, given AB =
10, BE = 7, and DE = 4
AD =
DB =
EF =
AF =
FC =
AC =
EC =
DF =
BC =
B
A
C
D
E
F
10
7
4
5
5
5
4
4
8
7
7
14
EXAMPLE 2 Solve for x if the perimeter of triangle ABC
is 42
B
A
C
D
E
F
12
3x – 3
9
12 9 (3 3) (3 3) 42x x
3x - 3
3x - 3
15 6 42x
6 27x
4.5x
HOMEWORK P. 288 #’s 10-18 even, 19-27, 29-32, 37,
40-44, 49-52
5.2 PERPENDICULAR
AND ANGLE BISECTORS
THEOREM 5-2: PERPENDICULAR BISECTOR THEOREM
Theorem If Then
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
PM AB and
MA MB
45555555555555 5PA PB
A BM
P
A BM
P
THEOREM 5-3: CONVERSE OF THE PERPENDICULAR BISECTOR THEOREM
Theorem If Then
If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.
PM AB and
MA MB
45555555555555 5PA PB
A B
P
A B
P
M
EXAMPLE 1 Calculate AC
A
B CD
6x – 104x
4 6 10x x
2 10x
5x
5x
6(5) 10AC
30 10AC 20AC
Complete Got it 1, p. 293
Ans: QR = 8
DISTANCE The distance from a point to a line is the
length of the perpendicular segment from the point to the line.
B
A
l
THEOREM 5-4: ANGLE BISECTOR THEOREM
Theorem If Then
If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
bisects ,
,
QS PQR
SP QP and
SR QR
55555555555555
55555555555555
55555555555555
SP SR
Q
P
S
R
Q
P
S
R
THEOREM 5-5: CONVERSE OF THE ANGLE BISECTOR THEOREM
Theorem If Then
If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.
,
,
SP QP
SR QR and
SP SR
55555555555555
55555555555555 bisects QS PQR
55555555555555
Q
P
S
R
Q
P
S
R
EXAMPLE 2 Calculate RM
7 2 25x x
Complete Got it 3, p. 295
Ans: FB = 21
A
R
P
M7x
2x+25
5 25x
5x
7(5)RM
35RM
HOMEWORK P. 296 #’s 6-8, 12-22, 36, 39-46
5.3 BISECTORS IN TRIANGLES
THEOREM 5-6: CONCURRENCY OF PERPENDICULAR BISECTOR THEOREM
Theorem Diagram Symbols
Perpendicular bisectors
are concurrent at P.
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices
, , PX PY and PZ
P
Z
YX
A
B C
PA PB PC
CONCURRENT, POINT OF CONCURRENCY, CIRCUMCENTER OF THE TRIANGLE, When 3 or more lines intersect at one point,
they are concurrent. The point at which they intersect is the point of concurrency.
The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.
CIRCUMSCRIBED ABOUT
Since the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of the circle that contains each vertex of the triangle. You say the circle is circumscribed about the triangle.
P
Z
YX
A
B C
EXAMPLE 1 Find the circumcenter of the following 3
points.
Circumcenter
THEOREM 5-7: CONCURRENCY OF ANGLE BISECTOR THEOREM
Theorem Diagram Symbols
Angle bisectors
are concurrent at P.
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.
, , AP BP and CP
PX PY PZ X
Z
YP
A
B C
INCENTER OF A CIRCLEINSCRIBED IN The point of concurrency of the angle
bisectors of a triangle is called the incenter of the triangle.
P is the center of the circle that is inscribed in the triangle.
X
Z
YP
A
B C
EXAMPLE 2 If P is the incenter and PY = 2x-4 and PZ
= 5x-13, find BP, if PB = 3x.
X
Z
YP
A
B C
2x-4
5x-133x
2 4 5 13x x
9 3x
3 x
3(3)PB
9PB
HOMEWORK P. 305 #’s 8-12 even, 15-18, 22, 33-34,
37-40
5.4 MEDIANS AND ALTITUDES
MEDIAN OF A TRIANGLE A median of a triangle is a segment
whose endpoints are a vertex and the midpoint of the opposite side.
J
D
F
E
Median
THEOREM 5-8: CONCURRENCY OF MEDIANS THEOREM
Theorem Diagram Symbols
The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
2
32
32
3
DC DJ
EC EG
FC FH
C
H
J
G
D
F
E
CENTROID OF THE TRIANGLE In a triangle, the point of concurrency of the
medians is the centroid of the triangle. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance. For any triangle, the centriod is always inside the triangle.
EXAMPLE 1 In the diagram below CE = 20, CH = 6,
and DC = 8. Find CG, FC and DJ.2
3CE GE
CG
H
J
D
F
E
220
3GE
3 2 320
2 3 2GE
30GE
30 20 10CG
206
8
10
EXAMPLE 1 In the diagram below CE = 20, CH = 6,
and DC = 8. Find CG, FC and DJ.1
3CH FH
CG
H
J
D
F
E
16
3FH
3 1 36
1 3 1FH
18FH
18 6 12FC
206
8
10
12
EXAMPLE 1 In the diagram below CE = 20, CH = 6,
and DC = 8. Find CG, FC and DJ.2
3DC DJ
Complete Got it 1, p. 310
Ans: a) ZC = 13.5 b) 2 : 1
CG
H
J
D
F
E
28
3DJ
3 2 38
2 3 2DJ
12DJ
206
8
10
12
ALTITUDE OF A TRIANGLE An altitude of a triangle is the
perpendicular segment from the vertex of the triangle to the line containing the opposite side. An altitude of a triangle can be inside or outside the triangle, or it can be a side of the triangle.
D
FE
FE
D
FE
D
THEOREM 5-9: CONCURRENCY OF ALTITUDES THEOREM
Theorem Diagram
The lines that contain the altitudes of a triangle are concurrent.
P
F
E
D
ORTHOCENTER OF THE TRIANGLE The lines that contain the altitudes of a
triangle are concurrent at the orthocenter of the triangle. The orthocenter can be inside, on, or outside the triangle.
F
E
D
Acute Triangle
F
E
D
Right Triangle
F
E
D
Obtuse Triangle
EXAMPLE 2 Triangle ABC has vertices A (1,3), B (2,7), and C (6,3).
What are the coordinates of the orthocenter of the triangle?
Graph the points
Create two altitudes
Locate the point of intersection
The orthocenter is the point (2, 4)
Complete Got it 3, p. 311Ans: (1, 2)
REVIEW OF THE 4 CONCURRENT POINTS
HOMEWORK P. 312 #’s 8-20, 24-27, 31, 37-41, 43-44
5.6 INEQUALITIES IN
ONE TRIANGLE
PROPERTY
Comparison Property of Inequality
If and 0, then a b c c a b
b c
a
COROLLARY TO THE TRIANGLE EXTERIOR ANGLE THEOREM
Corollary If Then
The measure of an exterior angle of a triangle is greater than the measure of each of the remote interior angles
and1
1
2
3
m
m
m
m
is an exterior a1 ngle
1
2
3
THEOREM 5-10 & THEOREM 5-11
Theorem 5-10 If Then
If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.
m mY Z
Theorem 5-11 If Then
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.
BC AC
XZ XY
X Z
Y
m mA B C
BA
EXAMPLE 1 In the diagram, list the sides in order
from smallest to largest.
Complete Got it 3, p. 327
Ans: OX
C
B
A
30°
110° ,AC ,BC
180 (30 110)m A 40m A
BA
THEOREM 5-12: TRIANGLE INEQUALITY THEOREM
Theorem Diagram
The sum of the lengths of any two sides of a triangle is greater than the third side.
Y
X
ZXY XYZ Z YZ XXZ Y XZ YXY Z
EXAMPLE 2 Can a triangle have side lengths of 4, 4,
and 8? How about 4, 5, and 6?
Complete Got it 4, p. 327Ans: a) No b) Yes
4 4 8
8 8
No, a triangle cannot be made with side lengths of 4, 4, and 8.
4 5 6
9 6
4 6 5
10 5
6 5 4
11 4
Yes, a triangle can be made with side lengths of 4, 5, and 6. The sum of the lengths of any two sides is greater than the third.
EXAMPLE 3 If two sides of a triangle are 8 and 14, what is
the range of possible lengths for the third side?
Complete Got it 5, p. 328
Ans: 3 < X < 11
8 14 x
22 x
8 14x
6x
14 8x
6x
The length of a side of a triangle can never be negative so we can eliminate that answer. So the range of values is between 6 and 22.
6 22x
HOMEWORK P. 328 #’s 6-34 even, 37-39, 43, 44, 46-
49, 53-55.
5.7 INEQUALITIES IN TWO TRIANGLES
THEOREM 5-13, THE HINGE THEOREM
Theorem 5-13 If Then
If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third leg is opposite the larger included angle.
m mA X
A
B
C
X Z
Y
BC YZ
THEOREM 5-14, CONVERSE OF THE HINGE THEOREM
Theorem 5-14 If Then
If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.
m mA X
A
B
C
X Z
Y
BC YZ
EXAMPLE 1 What is the range of possible values for
x?
Complete Got it 3, p. 335
Ans: -6 < x < 24
60 5 20x
U T
S
R
60°
15
10
(5x-20)°
Find the upper limit
80 5x16 x
Find the lower limit5 20 0x
5 20x 4x
4 16x
HOMEWORK P. 336 #’s 6-9, 11-14, 16-18, 21, 24, 26-
28, 31-35, 37-39