Download Journal 5 christian aycinena

Properties and Attributes of
Triangles
Chapter 5 Journal
Christian Aycinena 9-5
First of all, we need to know what is equidistant, or a point that is
the same distance from two or more objects.
Perpendicular Bisector:
A line that bisect a segment and is perpendicular to that segment.
Theorem:
Any point that lies on the perpendicular bisector is equidistant to
both of the endpoints of the segments.
Converse:
If it is equidistant from both of the endpoints of the segment, then
it is on the perpendicular bisector.
3cm
E
E
1 cm
We know that CD is perpendicular to
AB and that AO=OB, so using the
perpendicular bisector theorem we
know that
CB=AC(So if CB=3,then AC=3),
also that EB=AE(If EB=1, then AE=1)
finally AD =DB
We know that AC=CB, EA=EB and
AD=DB, so using the converse of
the perpendicular bisector theorem
we could prove by each segment
that CD is perpendicular to AB and
also that AO=OB
Angle Bisector:
A ray or line that cuts and angle into two congruent angles. It always
lies on the interior of the angle.
Theorem:
Any point that lies on the angle bisector is equidistant to both of the
sides of the angle.
Converse:
States that if a point is in the interior of an angle is equidistant from
the sides of the angle then it is on the angle bisector.
Ex. 2
S
A
C
X
4cm
P
Y
E
1.5cm
T
By this we know that
PS=PT=1.5(both), also XY=YZ and
CE=ED=4
Z
D
Ex. 3
From
daily Life.
Concurrent means a point where three or more lines intersect. The point of
concurrency is the point where they intersect.
Point of Concurrency
The point of concurrency where the three perpendicular bisectors of a triangle meet is
called the circumcenter of the triangle
The circumcenter is equidistant to the three vertices of the triangle.
MORE: Circumscribed: every vertex of the polygon(triangle in this case) lies on the circle
Ex. 1.
This is an example of
Circumcenter.
Ex. 3- Real Life
Ex. 2.
Ex. 4
The circumcenter of triangle
ABC is the center of the
circumscribed circle.
More Examples:
In an acute triangle circumcenter is on the inside of the
triangle
On a right triangle it is on the midpoint of the hypotenuse.
In an obtuse it is outside the triangle.
Acute Triangle
Circumcenter
Right Triangle Circumcenter
Obtuse Triangle Circumcenter
A triangle has three angle bisector, the angle bisector are
also concurrent and the point where the angle bisectors
of a triangle intersect is the
incenter.
=
Always occurs on the inside of the triangle.
Incenter is equidistant to any of the sides of the triangle.
Inscribed: A circle in which each side of the polygon is
tangent to the circle.
Ex. 1. The incenter
inscribes a circle in a
triangle and touches the
three sides.
Ex. 2
Ex. 3
Ex 4
A median of a triangle is a segment that goes from the vertex of a triangle to the
opposite midpoint. Every triangle has three medians and they are concurrent.
A
Median
Centroid
B
D
C
The centroid is a point of concurrency of the medians of a triangle.
The centroid is always inside the triangle and is also called the center of gravity.
The concurrency of medians of a triangle is located at 2/3 of the distance from each
vertex to the midpoint of the opposite side.
BG=2/3BF
AG=2/3AD
CG=2/3CE
Ex. 1.
Ex. 3.
Ex. 2Real life.
Altitude is a perpendicular segment from a vertex to the line
containing the opposite side of the triangle.
The point in which altitudes meet is called the orthocenter.
•If the triangle is acute the orthocenter is on the inside
of the triangle.
•If it is right triangle the orthocenter is on the vertex of
the right angle.
•If it is obtuse triangle it is on the outside of the
triangle.
Is a segment that joins the midpoints of two sides of the triangle.
Every Triangle has three mid segments which form the mid segment triangle.
Midsegments: AB, BC, CA
Midsegment triangle: ABC
The mid segment theorem says that a mid segment of a triangle is parallel to a
side of the triangle, and its length is half the length of its parallel side.
Ex. 1, Shows DE as a
midesegment, so
DE is parallel to BC and
DE=1/2BC
Ex. 2, Shows the
midesegment.
Ex. 3 A real life.
In any triangle the longest side is always opposite the largest
angle. The second largest side is always opposite the second
largest angle. The shortest side is opposite the smallest
angle.
In a triangle the largest angle is always opposite the longest
side. The second largest side is always opposite the second
longest side. The smallest angles is opposite the shortest
side.
A
Z
4.2cm
4.5cm
Y
B
Ex. 1. Angle R is the smallest
since it is in front of the
smallest side and angle Q is
the largest angle since it is in
front of largest side.
C
Ex.2. Angle C is the largest angle
since it is in front of largest side
Angle B is the smallest since it is
in front of the shortest side
Ex.3-Real life. Angle Z is the
largest angle since it is in
front of largest side Angle X
is the smallest since it is in
front of the shortest side
70 Y
30
Z
Ex. 1: Side CB is the largest since
it is in front of the larger angle
and side AB is the smallest since
it is in front of the smallest angle
Ex. 2. Side AB is the
smallest since it is in front
of the smallest angle. Side
AC is the largest since it is
in front of the larger angle
X
7 cm
80
X
Side ZY is the largest since it is in
front of the larger angle and side YX
is the smallest since it is in front of
the smallest angle
The exterior angle in a triangle is bigger than either of the non
adjacent interior angles.
Ex 1: W is an
exterior angle in this
triangle. Ands is
greater than <X and
greater than <Y.
In the figure,
exterior angle.
So,
is an
and
Angle OXZ is an
exterior angle, so it is
greater than XYZ and
XZY.
The sum of any two side lengths of a triangle is greater
that the third side length.
A
AB+BC>AC
BC+AB>AB
AC+AB>BC
B
Tell weather a triangle can have sides
with this lengths:
3,5,7-YES
8, 8,10-YES
9,1,5-NO
C
It is used when it is not possible to prove something directly.
Writing one:
1. Assume that what you are proving is FALSE.
2. Use that as your given, and start proving it.
3. When you come to a contradiction you have prove that it is
true
K
EX:
Triangle JKL is a right triangle
Prove: Triangle JKl does not have an obtuse angle.
1. Assume Triangle JKL has an obtuse angle. Let <k be obtuse.-Negation}
J
2. m<K +m<L =90—Acute <´s of a right triangle are comp.
3. m<k= 90- m<L –Subtraction Prop. Of =
4. m<k > 90– Def of obtuse <
5. _90-m<L >90 Substitute 90 – m<L for m<K
6. m<l < 0—Substract 90 from both sides and solve for m<L
By the Protractor Postulate, a triangle can´t have an angle with measure less than 0.
Therefore triangle JKL does not have an obtuse angle.
L
Examples #1
Triangle LMN has at most one right angle.
Step 1: Assume Triangle LMN has more than one right angle. That is, assume that angle L and
angle M are both right angles.
Step 2: If M and N are both right angles, then m∠L = m∠M = 90
Step 3: m∠L + m∠M + m∠N = 180 [The sum of the measures of the angles of a triangle is 180.]
Step 4: Substitution gives 90 + 90 + m∠N = 180.
Step 5: Solving gives m∠N = 0.
Step 6: This means that there is no Triangle LMN, which contradicts the given statement.
Step 7: So, the assumption that ∠L and ∠M are both right angles must be false.
Step 8: Therefore, Triangle LMN has at most one right angle.
Example #2
The supplement of an acute angle cannot be an acute angle.
Step 1: The supplement of an acute angle can be acute-NEGATION
Step 2: Supplementary angles add up to 180- Definition of supp.
Step 3: Acute angles measure less than 90-Definition of Acute <´s
Step 4: The sume of “ acute angles is less than 180-Sum Add Post.
5. Since this is our contradiction the supplement of an acute can´t be acute.
Theorem:
If two sides of one triangle are congruent to two sides of
another triangle and the included angles are not
congruent, then the longer third side across form the
larger included angle.
Converse:
If two sides of one triangle are congruent to two sides of
another triangle and the third sides are not congruent,
then the lager included angle is across form the longer
third side.
AC
EF
89
63
45
90
AC < EF
ZX
20
28
A
Y
m<Y
<
m<A
<
LM
5.5
5.7
M
F
m<F
>
m<M
In a 45-45-90 triangle, both legs are congruent, and the length of the
hypotenuse is the length of a leg times square root of 2
8
16
X
x
So you divide by
the square root of
two and X=32
X= 8 times
the square
root of 2
In a 30-6-90 triangle, the length of the hypotenuse is 2 times
the length of the shorter leg, and the length of the longer leg is
the length of the shorter leg times square root of 3
8
Y
X
X= 4
Y= 4 times
square root
of 3
Geometry Journal Chapter 5 Name Christian Aycinenna
In your own wordsrespond to the following:
_____(0-10 pts.) Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and
its converse. Give 3 examples of each.
_____(0-10 pts.) Describe what an angle bisector is. Explain the angle bisector theorem and its converse.
Give at least 3 examples of each.
_____(0-10 pts.) Describe what concurrent means. Explain the concurrency of Perpendicular bisectors of a
triangle theorem. Explain what a circumcenter is. Give at least 3 examples of each.
_____(0-10 pts.) Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter
is. Give at least 3 examples of each.
_____(0-10 pts.) Describe what a median is. Explain what a centroid is. Explain the concurrency of medians
of a triangle theorem. Give at least 3 examples of each.
_____(0-10 pts.) Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the
concurrency of altitudes of a triangle theorem. Give at least 3 examples.
_____(0-10 pts.) Describe what a midsegment is. Explain the midsegment theorem. Give at least 3
examples.
_____(0-10 pts.) Describe the relationship between the longer and shorter sides of a triangle and their
opposite angles. Give at least 3 examples.
_____(0-10 pts.) Describe the exterior angle inequality. Give at least 3 examples.
_____(0-10 pts.) Describe the triangle inequality. Give at least 3 examples.
_____(0-10 pts.) Describe how to write an indirect proof. Give at least 3 examples.
_____(0-10 pts.) Describe the hinge theorem and its converse. Give at least 3 examples.
_____(0-10 pts.) Describe the special relationships in the special right triangles (30-60-90 and 45-45-90).
Give at least 3 examples of each.
_____(0-5 pts.) Neatness and originality bonus.
______Total points earned (120 possible)