Transcript Bisectors of Triangles

Angle Bisectors and
Perpendicular Bisectors
Section 5.6
Objectives
• Use angle bisectors and perpendicular bisectors.
Key Vocabulary
•
•
•
•
Distance from a point to a line.
Equidistant
Angle bisector (review)
Perpendicular bisector
Theorems
• 5.3 Angle Bisector Theorem
• 5.4 Perpendicular Bisector Theorem
Angle Bisector
• A bisector which we have already studied is an  bisector.
• Definition: Angle Bisector - A ray that divides an angle into two
congruent angles.
In the picture to the right, the
red line segment is the angle
bisector. The  arc marks show
the 2  angles that were formed
when the angle bisector bisected
the original angle.
5
Angle Bisectors
• An  bisector divides an  into two ≅ parts.
• In a ∆, an  bisector divides one of the ∆’s s into two ≅
s. (i.e. if AD is an  bisector then BAD ≅ CAD)
Angle Bisector
Angle Bisectors
• Review – Distance
means ⊥ distance.
• Distance from a point
to a line is the length of
the  segment from the
point to the line.
• PD=distance from P to
line l.
P
D
l
Angle Bisectors
• When a point is the same distance from two or more
objects, the point is said to be equidistant from the
objects. Triangle congruence theorems can be used
to prove theorems about equidistant points.
• A locus is a set of points that satisfies a given
condition. The perpendicular bisector of a segment
can be defined as the locus of points in a plane that
are equidistant from the endpoints of the segment.
Angle Bisectors
• The bisector of an angle can be described as the
locus of points in the interior of the angle equidistant
from the sides of the angle.
B
A
D
C
• This description leads to the following theorems.
Angle Bisectors
Theorem 5.3 – Angle Bisector Theorem
• If a pt is on the bisector of an ,
then it is equidistant from the
sides of the .
D
A
O
Example: If OA bisects DOG, DA  OD,
and GA  OG, then DA  GA.
G
Example 1
Use the Angle Bisector Theorem
Prove that ∆TWU  ∆VWU.
UW bisects TUV.
∆UTW and ∆UVW are right triangles.
∆TWU  ∆VWU.
SOLUTION
Statements
1. UW bisects TUV.
2. ∆UTW and ∆UVW
are right triangles.
3. WU  WU
4. WV  WT
5. ∆TWU  ∆VWU
Reasons
1. Given
2. Given
3. Reflexive Prop. of Congruence
4. Angle Bisector Theorem
5. HL Congruence Theorem
Example 2: Applying the Angle
Bisector Theorem
Find the measure.
BC
BC = DC
 Bisector Thm.
BC = 7.2
Substitute 7.2 for DC.
Your Turn:
Given that YW bisects XYZ and
WZ = 3.05, find WX.
WX = WZ
 Bisector Thm.
WX = 3.05
Substitute 3.05 for WZ.
So WX = 3.05
Your Turn:
Find the measure of SR.
A. 22
B. 5.5
C. 11
D. 2.25
Your Turn:
C. Find the measure of UV.
A. 7
B. 14
C. 19
D. 25
Example 3: Using Angle Bisectors
Roof Trusses: Some roofs are built with wooden trusses that
are assembled in a factory and shipped to the building site. In
the diagram of the roof trusses shown, you are given that AB
bisects CAD and that ACB and ADB are right angles. What
can you say about BC and BD?
C A D
O
M
L
G
B
H
K
N
P
SOLUTION:
Because BC and BD meet AC and AD at right angles, they are
perpendicular segments to the sides of CAD. This implies that
their lengths represent distances from the point B to AC and AD.
Because point B is on the bisector of CAD, it is equidistant from
the sides of the angle.
So, BC = BD, and you can conclude that BC ≅ BD.
C A D
O
M
L
G
B
H
K
N
P
Review Definitions
• Midpoint: The midpoint of a segment is the point
halfway between the endpoints of the segment. If X
is the midpoint of AB, then AX = XB.
• Example:
A
X
B
• Segment Bisector: Any segment, line, or plane that
intersects a segment at its midpoint is called a
T
segment bisector.
• Example:
S
E
O
V
Perpendicular Bisector
• A segment, ray, line, or
plane that is
perpendicular to a
segment at its midpoint
is called a perpendicular
bisector.
midpoint
C
Given segment
A
CP is a  bisector of AB
P
perpendicular bisector
B
Perpendicular Bisector
Theorem 5.4 Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a
segment, then it is equidistant from the
A
endpoints of the segment.
C
D
Example: If AB is the perpendicular bisector of CD,
then AC  AD and BC  BD.
B
Example 4
Use Perpendicular Bisectors
Use the diagram to find AB.
SOLUTION
In the diagram, AC is the
perpendicular bisector of DB.
8x = 5x +12
By the Perpendicular Bisector
Theorem, AB = AD.
3x = 12
3x 12
=
3
2
x=4
Subtract 5x from each side.
Divide each side by 3.
Simplify.
You are asked to find AB, not just the value of x.
ANSWER
AB = 8x = 8 · 4 = 32
Triangle Perpendicular
Bisector
A ⊥ bisector of a ∆ is
a line, segment, or
ray that passes
through the midpoint
of one of the sides of
the ∆ at a right angle.
C
A
P
B
Side AB
perpendicular bisector
of ∆ABC
Triangle Perpendicular
Bisector
• The perpendicular bisector of a side of a triangle does
not necessarily pass through a vertex of the triangle.
Example: In ∆ABC,
the red line
segment is the ⊥
bisector side AC and
does not pass
through vertex B.
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Triangle Perpendicular
Bisector
Properties:
•Any point on the
perpendicular bisector
of a segment is
equidistant
from the endpoints of
the segment.
•Any point equidistant
from the endpoints of
a segment lies on the
perpendicular
bisector of the
segment.
Both sides are
congruent- make sure
you see this or it is
NOT a perpendicular
bisector
Perpendicular Bisector
Triangle Perpendicular
Bisector
Tell whether each red segment is an
perpendicular bisector of the triangle.
Example 5: Applying the Perpendicular
Bisector Theorem
Find each measure.
MN
MN = LN
 Bisector Thm.
MN = 2.6
Substitute 2.6 for LN.
Example 6: Applying the
Perpendicular Bisector Theorem
Find each measure.
TU
TU = UV
3x + 9 = 7x – 17
9 = 4x – 17
26 = 4x
6.5 = x
So TU = 3(6.5) + 9 = 28.5
 Bisector Thm
Substitute the given values
Subtract 3x from both sides
Add 17 to both sides
Divide both sides by 4
Your Turn:
Find the measure.
Given that line ℓ is the perpendicular bisector of DE and
EG = 14.6, find DG.
DG = EG
 Bisector Thm.
DG = 14.6
Substitute 14.6 for EG.
Your Turn:
A. Find the measure of NO.
A. 4.6
B. 9.2
C. 18.4
D. 36.8
Concurrent Lines
When two lines intersect at one point, we say that the lines
are intersecting. The point at which they intersect is
the point of intersection.
(nothing new right?)
Well, if three or more lines intersect at a
common point, we say that the lines are
concurrent lines. The point at which these
lines intersect is called the point of
concurrency.
Definitions
• Concurrent Lines – Three or more lines that intersect
at a common point.
• Point of Concurrency – The point where concurrent
lines intersect.
Perpendicular Bisector
and Concurrent Lines
• A triangle has 3 sides, so it
also has 3 perpendicular
bisectors, one for each side.
• The 3 perpendicular
bisectors are concurrent
lines.
• The point of concurrency of
the perpendicular bisectors
is called the circumcenter of
the triangle.
In the picture to the below
point K is the circumcenter.
Circumcenter
• The point of concurrency
of the three perpendicular
bisectors of a triangle is the
circumcenter.
• The circumcenter of a
triangle is one of many
different “centers” of a
triangle.
Circumcenter
Circumcenter
The circumcenter is
equidistant from all
three Vertices.
Circumcenter
•The circumcenter
gets its name
from the fact that
it is the center of
the circle that
circumscribes
the triangle.
• Circumscribe
means to be
drawn around a
figure and
passing through
each vertex.
Circumcenter
To find the center of a circle that will circumscribe any given
triangle, find the point of concurrency of the three perpendicular
bisectors of the triangle, the circumcenter. Sometimes this will be
inside the triangle, sometimes it will be on the triangle, and
sometimes it will be outside of the triangle!
Circumcenter
Acute ∆ - inside
Circumcenter
Right ∆ - on
Obtuse ∆ - outside
Circumcenter
• The Perpendicular Bisectors of
a triangle intersect at a point
that is equidistant from the
vertices of a triangle.
• Example: If P is the
circumcenter of ∆ABC, then
PA = PB = PC.
Example 7
Use Intersecting Bisectors of a Triangle
A company plans to build a warehouse
that is equidistant from each of its three
stores, A, B, and C. Where should the
warehouse be built?
SOLUTION
Think of the stores as the vertices of a triangle. The
point where the perpendicular bisectors intersect
will be equidistant from each store.
1. Trace the location of the stores on
a piece of paper. Connect the
points of the locations to form
∆ABC.
Example 7
Use Intersecting Bisectors of a Triangle
2. Draw the perpendicular bisectors
of AB, BC, and CA. Label the
intersection of the bisectors P.
ANSWER
Because P is equidistant from each vertex
of ∆ABC, the warehouse should be built
near location P.
Your Turn:
3 principals have a meeting. They want to meet at a
central location so everyone travels the exact same
distance.
1. How would you find
the central location?
1. Find the circumcenter
2. Why?
2. Same distance from the
3 vertices.
PRACTICE
Use Angle Bisectors and Perpendicular
Bisectors
1. Find FH.
ANSWER
5
ANSWER
20
ANSWER
15
2. Find MK.
3. Find EF.
Practice
No, right ∠ not indicated.
Practice
No, right ∠ not indicated.
Practice
Yes, right ∠ and midpoint
are indicated.
Practice
No, equidistant of
Point P not indicated.
Practice
No, segments from sides
to point P not indicated
as perpendicular.
Practice
m∠1=55
m∠2=55
m∠3=20
m∠4=70
m∠5=35
m∠6=50
Practice
m∠1=25
m∠2=65
m∠3=65
m∠4=115
m∠5=115
m∠6=20
m∠7=45
m∠8=45
Joke Time
• How can you get four suits for a dollar?
• Buy a deck of cards.
• Guy buys a parrot that is constantly using foul language. Really
horrible stuff. Finally the guy gets fed up and throws the
parrot in the freezer to punish him. After about an hour, he
hears a faint tapping sound from inside the freezer and opens
the door. There’s the parrot, wings wrapped around himself,
shivering. He says,
• “I swear, I’ll never, ever curse again. But can I ask you a
question? What did the chicken do?"
Assignment
• Pg. 276 – 280: #1 – 25 odd, 33 – 37 odd, 33, 35, 45,
47