CHAPTER 5
Relationships within
Triangles
By Zachary Chin and
Hyunsoo Kim
THIS is a triangle.
For those of you who are unaware…
How to draw a Triangle
1
2
3
Step 1: Draw a line segment.
Step 2: Draw another line segment with a
common endpoint.
Step 3: Connect with another line segment.
NO
DUH!
5.1: Midsegments and
Coordinate Proofs
Midsegments
A midsegment is a line in a triangle that
connects the midpoints of two sides of a
triangle. Every triangle has three
midsegments.
Midsegment Theorem
The segment connecting the midpoints of
two sides of a triangle is parallel to the third
side and is half as long as that third side.
B
D
A
>
>
DE AC and
DE=½AC
E
C
Example #1
Find the length of the
midsegment (DE is a
midsegment)
A
20
E
DE = ½AC, so…
DE = 10
B
D
C
Coordinate Proofs
A coordinate proof involves placing
geometric figures in a coordinate plane.
When you use variables to represent the
coordinates of a figure in a coordinate
proof, the results are true for all figures of
that type.
Coordinate Proofs
Triangle:
y
(0, 0)
(a, b)
(c, 0)
x
Example #2
Place a regular octagon in the coordinate plane.
y
(b, f)
(c, f)
(0, e)
(d, e)
(0, a)
(d, a)
(b, 0)
(c, 0)
x
5.2: Use Perpendicular
Bisectors
A segment, ray, line, or plane that is
perpendicular to a segment at its midpoint is
called a perpendicular bisector.
A point is equidistant from two figures if the
point is the same distance from each figure.
Perpendicular Bisector
Theorem
In a plane, if a point is on the perpendicular
bisector of a segment, then it is equidistant
from the endpoints of the segment.
C
A
P
B
If CP is the perpendicular bisector of AB, then CA = CB.
Converse of the Perpendicular
Bisector Theorem
In a plane, if a point is equidistant from the endpoints
of a segment, then it is on the perpendicular bisector
of the segment.
C
A
P
B
D
If DA = DB, then D lies on the perpendicular bisector
of AB.
Example #3
C
-6x+4
10x2+15x-6
A
P
B
CP is the perpendicular bisector of AB. Find CB.
Use CA = CB, so…
10x+15x-6 = -6x+4
Solve for x, plug it into -6x+4…
CB = 1.6 units
When three or more lines, rays, or
segments intersect in the same point, they
are called concurrent lines, rays, or
segments. The point of intersection of the
lines, rays, or segments is called the point
of concurrency.
Concurrency of Perpendicular
Bisectors Theorem
The perpendicular bisectors of a triangle
intersect at a point that is equidistant from
the vertices of a triangle.
circumscribed circle
circumcenter
5.3: Use Angle Bisectors of
Triangles
Angle Bisector Theorem:
If a point is on the bisector of an angle, then it is equidistant
from the two sides of the angle.
Converse of the Angle Bisector Theorem:
If a point is in the interior of an angle and is equidistant from
the sides of the angle, then it lies on the bisector of the
angle.
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.
inscribed circle
incenter
Example #4
6x-4
3x+2
Find the value of x.
BD = CD, so…
6x-4 = 3x+2
x=2
5.4: Use Medians and
Altitudes
A median of a triangle is a segment from a
vertex to the midpoint of the opposite vertex.
The point of concurrency, called the centroid,
is inside the triangle.
Concurrency of Medians of
a Triangle
centroid
The medians of a triangle intersect at a point
that is two-thirds the distance from each vertex
to the midpoint of the opposite side.
For example, AG = 2/3 AMA
An altitude of a triangle is the perpendicular
segment from the vertex to the opposite side or to
the line that contains the opposite side.
Concurrency of Altitudes of
a Triangle
The lines containing the altitudes of a
triangle are concurrent.
orthocenter
Example #5
If AG = 2x+3, and AMA = 6x+6, find the value of x.
AG = 2/3 AMA
2x+3 = 2/3 (6x+6)
x = -½
5.5 Use Inequalities in a
Triangle
Theorem 5.10
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.
8
This angle is larger
than…
12
This one
Theorem 5.11
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.
A
C
AC < BC because m∠A is greater than m∠B.
B
Example #6
Given that m∠A is greater than m∠C, which
side is longer, AB or BC?
The answer is BC.
The Triangle Inequality
Not all triangles can be made with any
lengths. If you have a “triangle” that has
sides of 2, 2, and 4, this triangle is not
possible to draw. 4=4, as 2+2=4, so that
means that this triangle will become a
straight line. This “triangle inequality” leads
to…
Triangle Inequality Theorem
The sum of the lengths of any two sides of a
triangle is greater than the length of the
third side.
Example #7
Is it possible to make a triangle with sides of
43.6, 57.2, and 101.4?
No, because 100.8 (43.6+57.2) is less than 101.4.
5.6: Inequalities in Two
Triangles
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
triangle.
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 triangle is
longer than the third side of the second triangle, then the
included angle of the first triangle is larger than the
included angle of the second triangle.
Example #8
5
3
Given BC and EF, which is larger, ∠A or ∠D?
∠A
Because 5 > 3.