Chapter 5 – Properties of Triangles
The Bigger Picture
-Properties of Triangles such as perpendicular and angle
bisectors and how they relate in triangles
- Congruency of perpendicular and angle bisectors
- The use of medians and altitudes to locate points on a
triangle
The “What” and the “Why”
Use of properties of perpendicular
bisectors and angle bisectors
-Optimize the positioning of a goalie in hockey
or soccer
Use properties of perpendicular
bisectors and angle bisectors
- Using the concept of equidistant to locate
objects and points
Use properties of medians and
altitudes of a triangle
- Find points on a triangle and use them to
measure various objects such as a person’s
heart fitness
Use properties of mid-segments of a triangle
- Determining length
Compare the length of sides or the measures
of the angles of a triangle
-Determining balance points for structural projects such as
cranes and booms
Understand and write indirect proofs
-Prove theorems that cannot be easily proved directly
Use the Hinge Theorem and its converse to
compare side lengths and angle measures of
triangles.
- Comparative analysis tool for measuring distance to a
common point
Properties of Triangles
Soccer goalkeepers use triangle
relationships to help block shots
on goal.
An opponent can shoot the ball from
many different angles. The
goalkeeper determines the best
defensive position by imagining a
triangle formed by the goal posts
and the opponent.
The opponent x is trying to score a
goal. Which position do you think
is best for the goalkeeper, A, B, or
C? Why?
A
Estimate the measure of <X, known
as the shooting angle. How could
the opponent change positions to
improve the shooting angle?
C
B
Using Properties of Perpendicular Bisectors
In chapter 1 we learned that a segment bisector intersects a segment at its
midpoint. A segment, ray, line, or plane that is perpendicular to a segment at it’s
C
midpoint is called a Perpendicular Bisector.
CP is a _|_ bisector of AB
P
A
Theorem 5.1 Perpendicular Bisector Theorem
B
C
If a point is on the perpendicular bisector of a segment, then it is
equidistant from the endpoints of the segment.
A
P
B
If CP is the _|_ bisector of AB, then CA = CB
C
Theorem 5.2 Converse of the Perpendicular Bisector
Theorem
If 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
A
P
D
B
Using Perpendicular Bisectors
If we are told that MN is the perpendicular bisector of ST, how
can we use that to help determine other aspects about the
figure?
What segment lengths in the diagram are equal? Why?
Explain why Q is on MN.
T
12
M
N
Q
12
S
Using the Properties of Angle Bisectors
Consider the following: the distance from a point to a line is
defined as the length of the perpendicular segment from the point to
the line.
And, when a point is the same distance from one line as it is from
another, then the point is equidistant from the two lines.
Knowing that, we can then apply the logic to a point on the interior
of an angle, and use it to help determine the bisector of the angle.
Q
*P
P
m
Angle Bisector Theorems
Theorem 5.3 Angle Bisector Theorem:
If a point is on the bisector of an angle, then the point is
equidistant from the two sides of the angle.
If m<BAD = m<CAD, then DB = DC
B
A
D
C
Theorem 5.4 Converse of Angle Bisector
Theorem
B
If a point is in the interior of an angle and is equidistant
from the sides of the angle, the it lies on the bisector of
the angle.
A
D
If DB = DC, then m<BAD = m<CAD
C
Proof of Theorem 5.3 – Angle Bisector Theorem
B
Given: D is on the bisector of <BAC
DB _|_ AB, DC _|_ AC
Prove: DB = DC
~ /\ ADC, Then
Plan: Prove /\ ADB =
Conclude DB =~ DC, so DB = BC
D
A
C
Statements
Reasons
~ <CAD
1. <BAD =
1. Definition of an Angle Bisector
2. <ABD & <ACD are Right Angles 2. Definition of Perpendicular Line
~ <ACD
3. <ABD =
3. Right Angle Congruence Theorem
4. AD =~ AD
4. Reflexive Property of Congruence
~
5. /\ ADB = /\ ADC
5. AAS Congruence Theorem
6. BD ~
= DC
6. CPCTC
7. BD = DC
7. Definition of Congruent Segments
Bisectors of a Triangle
Perpendicular Bisector of a Triangle:
A line (ray of segment) that is perpendicular to the side of the triangle at it’s midpoint.
Concurrent Lines:
Three of more lines that intersect at the same point.
Point of Concurrency:
The point of intersection of the lines is called the point of concurrency.
The three perpendicular bisectors of a triangle are concurrent. And the
point of concurrency is known as the Circumcenter and can be located
inside, on, or outside the triangle.
Acute
Right
Obtuse
Theorem 5.5 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
The point of concurrency (P) of the perpendicular bisectors of the
triangle is called the Circumcenter of the Triangle
Where and how might this be helpful in the real world?
Perpendicular Bisectors of a Triangle and Facilities Planning – an example
Client F
A company is planning to
build a new distribution facility
that is convenient to all of it’s
major clients.
*
Client E
How might locating the
circumcenter of the three
clients be beneficial in
determining the location?
*
*
Client G
Using Angle Bisectors of a Triangle
An Angle Bisector of a triangle is a bisector of an angle of the
triangle.
The three angle bisectors are concurrent.
The point of concurrency is called the incenter of the triangle.
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 = PF
Using Angle Bisectors
P
R
S
Based upon the Angle
Bisector Theorem, which
segments are congruent?
L
M
Q
If the length of ML = 17, and the length of MQ = 15, can we
determine the length of LQ, LS, and LR?
N
Medians and Altitudes of Triangles
A
Median – a segment whose endpoints are a vertex and
the midpoint of the opposite side of the triangle.
The three Medians are Concurrent
E
The point of concurrency is called the Centroid of the
triangle
B
F
P
D
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 = 2/3CE
C
Altitude of a Triangle
An altitude of a triangle is a perpendicular segment from a
vertex to the opposite side of a triangle, or to a line that contains
the opposite side of the triangle. An altitude can lie inside,
outside, or on the triangle.
Altitudes are concurrent, and the point of concurrency is called
the orthocenter of the triangle.
Acute
Right
Obtuse
Summary – Triangle Bisectors
Perpendicular Bisector of a Triangle:
A line (ray of segment) that is perpendicular to the side of the triangle at it’s midpoint.
The point of concurrency is called the Circumcenter.
Acute
An Angle Bisector of a triangle is a
bisector of an angle of the triangle.
The point of .concurrency is called
the Incenter.
Median – a segment whose
endpoints are a vertex and the
midpoint of the opposite side of the
triangle.
The three Medians are Concurrent
and the point of concurrency is
called the Centroid of the triangle
Right
Obtuse
An altitude of a triangle
is a perpendicular
segment from a vertex to
the opposite side of a
triangle, or to a line that
contains the opposite side
of the triangle. An altitude
can lie inside, outside, or
on the triangle.
Altitudes are concurrent,
and the point of
concurrency is called the
orthocenter of the
triangle.
Mid-Segment Theorem
A Mid-segment of a triangle is a segment that connects the midpoints of two
sides of a triangle.
Theorem 5.9 Mid-segment Theorem
The segment connecting the midpoints of two sides of a triangle is
parallel to the third side, and is half as long.
C
DE || AB and DE = ½ AB
D
A
E
B
Using the Mid-segment Theorem
UW and VW are mid-segments of Triangle RST.
S
VW = 8, RS = 12.
Find UW and RT.
W
T
V
U
R
Proving the Mid-segment Theorem
1. DE is the mid-segment for sides AC
and BC.
2. Using the midpoint formula, we can
determine the coordinate values for
points D and E.
D = (5, 5)
E = (11, 5)
C(10, 10)
3. Determine the slope of DE
- Compare that to the slope of AB
4. Determine the length of DE and AB
using the distance formula.
- Since they are both horizontal lines,
the length can be determined as the
absolute value of the difference in the x
values.
DE = 6
AB = 12
5. DE || AB & DE = ½ AB
D
A(0, 0)
E
B (12, 0)
Using Midpoints to Draw a Triangle
The midpoints of the sides of a triangle
are L (4, 2), M(2, 3), and N(5, 4).
What are the coordinates of the vertices
of the triangles?
Perimeter of a Mid-segment Triangle
Given ST = 12, TR = 10, and
SR = 8,
S
What is the perimeter of Triangle
UVW?
W
T
V
U
R
Hinge Theorem
Theorem 5.14 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 triangle is longer than the third side of the second.
Theorem 5.15 Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of
another triangle, and the third side of the first is longer than the
third side of the second, then the included angle of the first is
larger than the included angle of the second.
Indirect Proof of Theorem 5.15 – Example 2
Finding Possible Side Lengths and Angle Measures
E
B
80
D
36
F
A
C
1.
__ __
__ __
~
~
If AB = DE and BC __
= EF, AC = 12, m<B = 36*, m<E = 80*. Which of the following is a
possible length for DF? 8, 10, 12, or 23?
2.
__ __ __ __
~
~
In a triangle RST and triangle XYZ, RT = XZ, ST = YZ, RS = 3.7, XY = 4.5, and m<Z = 75*.
Which of the following is a possible measure for <T? 60*, 75*, 90*, or 105*?
Comparing Distances using the Hinge Theorem
You and a friend are flying separate planes. You leave the airport
and fly 120 miles due west. You then change direction and fly
W 30* N for 70 miles. Your friend leaves the airport and flies 120
miles due east. She then changes direction and flies E40* S for 70
miles. Each of you have flown exactly 190 miles, but which one of
you is farther from the airport?
P
airport
Comparing Distances using the Hinge Theorem
1.
Your flight: 100 miles due west, then 50 miles N20* W.
Your Friend: 100 Miles due north, then 50 miles N30* E
2. Your flight: 210 miles due south, then 80 miles S70* W.
Your Friend: 80 miles due north, then 210 miles N50* E.
P
airport