5'1 Perpendiculars and Bisectors

1
5.1 Perpendiculars and Bisectors

• Day 1 Part 1
• CA Standard 16.0

2
Warmup

• Simplify.
• 1. 6x 11y 4x y
• 2. -5m 3q 4m q
• 3. -3q 4t 5t 2p
• 4. 9x 22y 18x 3y
• 5. 5x2 2xy 7x2 xy

3
Perpendicular Bisector Theorem

• 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.

C
B
P
A
4

• Converse of the Perpendicular Bisector Theorem

5
In the diagram shown, MN is the perpendicular
bisector of ST.

• What segment lengths in the diagram are equal?
• Explain why Q is on MN

T
12
Q
N
M
12
S
6
Angle Bisector Theorem

• If a point is on the bisector of an angle, then
  it is equidistant from the two sides of the
  angle.
• If mltBAD mltCAD, then DB DC.

B
D
A
C
7

• Converse of the Angle Bisector Theorem

8
Use the diagram to answer the following. In the
diagram, F is on the bisector of lt DAE.

• If mltBAF 50, then mltCAF ____
• If FC 10, then FB ____
• Is triangle ABF congruent to triangle ACF?
  Explain.

A
B
C
E
D
F
G
9
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10
5.2 Bisectors of a Triangle

• Day 1 Part 2
• CA Standards 16.0, 21.0

11

• In the figure, YW bisects ltXYZ.
• mltXYZ 6x 2, mltZYW 8x 6.
• Solve for x and find mltXYZ.

W
Z
X
Y
12
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.
• PA PB PC

B
P
C
A
P is also called the circumcenter of the triangle.
13
Use the diagram shown.

• E is the circumcenter of ? ABC.
• DA ___
• BF ___
• ltEFC ___

A
E
D
C
B
F
14
Definitions

• Concurrent lines three or more lines intersect
  in the same point.
• Point of concurrency the point of intersection
  of the lines.

15
Concurrency of Angle Bisectors of a Triangle

• The angle bisectors of a triangle intersect at a
  point that is equidistant from the vertices of
  the triangle.
• PD PE PF

B
D
F
P
C
A
E
16

• The point of concurrency can be inside the
  triangle, on the triangle, or outside of the
  triangle.
• Acute Triangle inside
• Right Triangle on
• Obtuse Triangle outside

17
Example
M

• Which segments
• are congruent?

Q
R
S
P
N
L
18
Use the diagram shown.

• E is the circumcenter of ? ABC.
• DA ___
• BF ___
• ltEFC ___

A
E
D
C
B
F
19
Mini quiz on definitions

• The _____________ of the angle bisectors is
  called the incenter of the triangle.
• If three or more lines intersect at the same
  point, the lines are ________.
• The point of concurrency of the perpendicular
  bisectors of a triangle is called
  ____________________.

Point of concurrency
Concurrent
Circumcenter of the triangle
20
Construction

• Pg. 268 14, 15
• Pg. 275 5 9

21

• Pg. 269 21 29, 32
• Pg. 275 10 - 22

22
5.3 Medians and Altitudes of a Triangle

• Day 2 Part 1
• CA Standards 16.0

23
Warmup

• Find BD.

C
D
15
B
A
12
12
24

• AC ___
• mltDCB ___
• mltB ___

20
55
35
A
D
L
B
55
20
C
25
Median of a triangle.

• Median of a triangle is a segment whose endpoints
  are a vertex of the triangle and the midpoint of
  the opposite side.

Median
26

• The three medians of a triangle are concurrent.
  The point of concurrency is called the centroid
  of the triangle.

Centroid
P

27
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.
• AP 2/3 AD
• BP 2/3 BF
• CP 2/3 CE

C
F
D
P
A
B
E
28

• P is the centroid of ?QRS shown. Find RT and RP
  when PT 5.

R
P
S
Q
T
29

• Sketch ?JKL with J(7,10), K(5,2), L(3,6).
• Find the coordinates of the centroid of ?JKL.

30
Altitude of a triangle

• An altitude of a triangle is the perpendicular
  segment from a vertex to the opposite side or to
  the line that contains the opposite side.

Altitude
31

• Every triangle has three altitudes.
• The lines containing the altitudes are concurrent
  and intersect at a point orthocenter of the
  triangle.
• Where is the orthocenter located in each type of
  triangle?
• Acute triangle
• Right triangle
• Obtuse triangle

32
Use the diagram shown and the given information
to decide in each case whether EG is a
perpendicular bisector, an angle bisector, a
median, or an altitude of ? DEF.
E

• DG FG
• EG DF
• mltDEG mltFEG

Median
T
Altitude
Angle bisector
D
F
G
33

• The angle bisectors of ? ABC meet at point D.
  Find DE.

19
B
F
E
L
L
D
19
28
C
A
G
34
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35
5.4 Midsegment Theorem

• Day 2 Part 2
• CA Standards 17.0

36
Review

• Given PQ 14, SU 6, and QU 3, find the
  perimeter of ? STU.

Q
S
U
P
R
T
37
Midsegment Theorem

• The segment connecting the midpoint of two sides
  of a triangle is parallel to the third side and
  is half as long.
• DE ll AB and DE ½ AB

C
D
E
gt
gt
B
A
38
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39
UW and VW are midsegment of ? RST. Find UW and
RT.
R
16
U
T
V
12
8
6
W
S
40
GH, HJ and JG are midsegments of ? EDF.

• JH ll ___
• EF ___
• DF ___
• ___ ll DE
• GH ___
• JH ___

24
DF
J
D
E
21.2
8
10.6
16
H
G
GH
12
8
F
41
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42

• Given the midpoints of a triangle are (7,4),
  (5,6) and (8,7), find the coordinates of the
  vertices.

43

• Pg. 282 3 12
• Pg. 283 17 20
• Pg. 290 3 22, 26 29

44
5.5 Inequalities in One Triangle

• Day 3 Part 1
• CA Standards 6.0, 13.0

45
Warmup

• Solve the inequality.
• 1. -x 2 gt 7
• 2. c 18 lt 10
• 3. -5 m lt 21
• x 5 gt 4
• z 6 gt -2

46
Theorems

• If one side of a triangle is longer than another
  side, then the angle opposite the longer side is
  larger than the angle opposite the shorter side.
• mltB gt mltC

A
3
7
B
C
47

• List the angles in order from greatest to least.

A
27
18
B
C
23
48

• If one angle of a triangle is larger than another
  angle, then the side opposite the larger angle is
  longer than the side opposite the smaller angle.

D
60
EF gt DF
E
40
F
49

• Write the measurements of the triangles in order
  from least to greatest.

J
Q
7
100
R
5
45
G
6
35
H
P
JG, HJ, HG
mltR, mltQ, mltP
50

• List the sides in order from longest to shortest.

F
65
45
G
E
51
Exterior Angle Inequality Theorem

• The measure of an exterior angle of a triangle is
  greater than the measure of either of the two
  nonadjacent (not next to) interior angles.

mlt 1 gt mltA mlt1 gt mltB
A
1
B
C
52
Constructing a Triangle

• Construct a triangle with the given group of side
  length, if possible.
• 4 in, 4 in, 4 in
• 2 in, 4 in, 6 in
• 3 in, 4 in, 5 in

53
Triangle Inequality Theorem

• The sum of the lengths of any two sides of a
  triangle is greater than the length of the third
  side.
• AB BC gt AC
• AC BC gt AB
• AB AC gt BC

C
A
B
54

• Use the diagram to solve the inequality
• AB BC gt AC.

B
4x 5
3x 2
C
6x 3
A
55

• Two sides of a triangle have lengths 4 and 14.
  Describe the possible length of the third side.
• Two sides of a triangle have lengths 10 and 15.
  Describe the possible length of the third side.

56
5.6 Indirect Proof and Inequalities in Two
Triangles

• Day 3 Part 2
• CA Standards 2.0, 6.0, 13.0

57
Review

• What is the sum of x and y?
• Which measure is greater, x or y ?

x
20
y
21
58
Hinge Theorem

• If two sides of one triangle are congruent to two
  sides of another triangle, and the included angle
  of the first is larger than the included angle of
  the second, then the third side of the first is
  longer than the third side of the second.

RT gt WX
V
S
80
100
T
X
W
R
59
Use Hinge theorem to complete the blank with lt,
gt, or .
lt
gt

• 1. RS __ TU 2. mlt1 ___ mlt 2
• 3. XY __ ZY

1
S
110
U
13
15
R
130
2
T
Z
lt
41
38
Y
X
60
Converse of the Hinge Theorem
61

• What is the largest angle that is part of a
  triangle?

W
9
9
10
Y
X
4
4
Z
62
Review

• List the sides in order from shortest to longest.

L
C
50
75
49
A
31
N
81
74
B
M
49 50 mltB 180 99 mltB 180 mltB
81
75 74 mltN 180 149 mltN 180 mltN 31
BC, AB, AC
LM, LN, MN
63
Review

• Solve for a

2a 7
a 19
64
Extra Credit!!

• Use the diagram to solve.
• Find the value of x.
• Find mltB
• Find mltC
• Find mltBAC

D
A
3x
(x13)
(x19)
C
B
65

• Pg. 298 6 25, 34
• Pg. 305 3 23, 26, 27

66
Ch 5 Review

• Day 4 Part 1

67
Warmup
68
Review

• Find the value of x.

40
2x6
x
69
Review
70

• Ch 5 Culminating Task

71
Ch 5 Test

• Day 4 Part 2