Chapter 5

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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 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

• 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 persons heart
  fitness

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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?
• Estimate the measure of ltX, known as the shooting
  angle. How could the opponent change positions
  to improve the shooting angle?

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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 its midpoint is called a
Perpendicular Bisector.
CP is a __ bisector of AB
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Using Perpendicular Bisectors
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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.
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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
mltBAD mltCAD, then DB DC Theorem 5.4
Converse of Angle Bisector Theorem 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. If DB DC, then mltBAD
mltCAD
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Proof of Theorem 5.3 Angle Bisector Theorem
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Bisectors of a Triangle
Perpendicular Bisector of a Triangle A line
(ray of segment) that is perpendicular to the
side of the triangle at its 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.
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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
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?
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Perpendicular Bisectors of a Triangle and
Facilities Planning an example

Client F

Client E




Client G

A company is planning to build a new distribution
facility that is convenient to all of its major
clients. How might locating the circumcenter of
the three clients be beneficial in determining
the location?
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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
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Using Angle Bisectors
Based upon the Angle Bisector Theorem, which
segments are congruent?
If the length of ML 17, and the length of MQ
15, can we determine the length of LQ, LS, and LR?
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Medians and Altitudes of Triangles
Median a segment whose endpoints are a vertex
and the midpoint of the opposite side of the
triangle. The three Medians are Concurrent The
point of concurrency is called the Centroid of
the triangle
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
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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.
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Summary Triangle Bisectors
Perpendicular Bisector of a Triangle A line
(ray of segment) that is perpendicular to the
side of the triangle at its midpoint. The point
of concurrency is called the Circumcenter.
An Angle Bisector of a triangle is a bisector of
an angle of the triangle. The point of
.concurrency is called the Incenter.
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.
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
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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. DE AB and DE ½ AB
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Using the Mid-segment Theorem
UW and VW are mid-segments of Triangle RST. VW
8, RS 12. Find UW and RT.
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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) 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
C(10, 10)
D E
A(0, 0)
B (12, 0)
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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?
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Perimeter of a Mid-segment Triangle
Given ST 12, TR 10, and SR 8, What is the
perimeter of Triangle UVW?
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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
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Finding Possible Side Lengths and Angle Measures
E
B 80

36 D
F
A C
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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
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Comparing Distances using the Hinge Theorem

• 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