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Transcript
By: Ana Julia Rogozinski
(YOLO)
- A perpendicular bisector is the division of a line
when making two congruent halves by passing
through its midpoint, creating right angles.
-EXPLANATION: When you have a segment and you
want to construct a perpendicular bisector your
first step would be to find the midpoint of the
segment, by using a compass. Then you draw a line
right between the middle and remember it has to
be perpendicular to the line meaning both sides will
be the same and will form angles of 90° (right
angles).
1. YA = YB
W
Y
A
2.WA=WB
B
3.EA= EB
E
-Perpendicular bisector theorem states: The
Perpendicular bisectors of the sides of a triangle are
concurrent at a point that is equidistant from the
vertices of the triangle
--If a point is on the perpendicular bisector of a
segment, then it is equidistant from the endpoints
of the segment.
-Converse: Is a point is equidistant from the
endpoints of a segment, then it is in the
perpendicular bisector of the segment.
A
1. CZ= ZB
2. AB= AC
B
C
Z
3. What can you prove by the converse of the
perpendicular bisector theorem?
You can prove XA= AN
N
X
A
-an angle bisector, it is a line passing through
the vertex of the angle that cuts it into two
equal smaller angles (in half). The bisector is
what is cutting the angle or triangle in half.
Y
X
W
2x+5
F
Example 3:
If X =2 then what is
the measurement of
YX? 9
-Angle bisector theorem: If there is a point bisecting
an angle then it is equidistant from the sides of the
angle.
-Converse: If the point inferior of an angle is
equidistant from the sides of the angle, then is is on
the bisector of the angle.
-- Concurrent means the three or more lines that
intersect all at a point.
- The point of concurrency is the point in which
all lines intersect.
- The concurrency of perpendicular bisector of
a triangle is when the three sides of the triangle
all have a perpendicular line bisecting them in
the middle and when those lines intersect they
are said to be concurrent.
It’s theorem states that the circumcenter of a
triangle is equidistant from the vertices of the
triangle.
Example 1
Example 2
Example 3
Example 1
Example 2
Example 3
-Circumcenter is the point where the three
perpendicular bisectors intersect. The circumcenter
can lie on the inside of the triangle and sometimes
even outside of it. Its theorem states that: a
triangle is equidistant from the vertices of the
triangle.
Example 1
Example 3
Example 2
An angle bisector of a triangle is when the three
angles of a triangle are bisected creating 2 equal
angles for each and when the bisecting lines that
intersect are concurrent, that point is the incenter
of a triangle.
It’s theorem states that the incenter of a triangle is
equidistant from the sides of the triangle.
- An incenter is the center of a triangles inscribed
circle.
-An incenter is equally distant from the sides of the
triangle.
Example 1
Example 2
Example 3
A median is a line in a triangle in which one end is in
the middle of one of its side and the other is the
vertex. Every triangle has three medians and when
they intersect they are concurrent.
-Medians intersect at the centroid.
-Theorem: The centroid of a triangle is 2/3 of
the distance from each vertex to the midpoint of
the opposite side
Example 1
Example 3
Example 2
- Altitudes intersect at the orthocenter.
- The altitude of a triangle is a line segment from
one vertex of a triangle to the opposite side so
that the line segment is PERPENDICULAR to the side.
-Some altitudes can even fall outside the triangle.
For example in obtuse triangle.
The point where the three altitudes of
a triangle intersect.
Example 1
Example 3
Example 2
Example 1
Example 3
Example 2
-- A midsegment is a segment that connects
the midpoints of two sides of a triangle; The
midsegment triangle is created since every
triangle has 3 midsegments. Three
midsegments intersect to form a triangle
with ½ the perimeter of the
larger triangle
-Theorem: A midsegment connecting two
sides of a triangle is parallel to the third side
and is half as long
Example 1
Example 3
Example 2
If AD=DB and AE = EC,
then DE || BC
X= 3 because PQ = ½ of BC
-When having only some measurements of the sides
and of some angles, in order for you to know all the
sides or angles in the triangle from the biggest to
smaller, you can use this.
- When having the angles and you want to see the
order of the sides for example, you would see the
opposite angle from each side and depending on the
measurement of the angles , you will see the
measurement of the sides. For example the bigger
angle on a triangle ,its opposite side will be the
bigger as well.
Example 1
e
30
What are the order of the angles from
the smallest to the biggest?
F, g, e
32
g
f
38
Example 2
R
78
M
What is the order of the sides from bigger to
smaller?
U, M , R
62
U
40
Example 3
Which angle is the
smalles? F
N
11.2
11
P
F
11.3
C
35
Example 1
Which is the longer side?
CF
F
61
54
W
Example 2:
Which is the shortest
side? FW
Example 3:
Order the sides from greatest
to shortest: CF , CW, FW
-Since we know from the exterior angle theorem that
the exterior angle is the sum of the 2 other interior
angles , we know then that the exterior angle must
be larger that either of the angles that need to be
added.
-An exterior angle of a triangle is greater than either
of the non-adjacent interior angles.
A
D
T
X
1. Which angle is greater <ADX or
< XDR? <ADX
2. Which angle is greater <DXR or
<TXD? < TXD
3. <QRD id greater that < DRX
R
Q
-The triangle inequality states that for any
triangle the sum of two of the lengths of any
two sides must be greater than the length of
the remaining side.
1. Can 2.23, 2.12 ,4 make a triangle?
YES, because the sum of 2.23 and 2.12 is greater than
4.
2. Can 3.5, 4.8, 8 make a triangle?
NO, because they do not add up to more than 8
3. What type of triangle do 9,14,17 create? They create
an obtuse triangle
-Indirect proofs are used when it is not possible to
prove something directly.
-Steps to write an indirect proof:
1st: State all possibilities
2nd:Assume that what you are proving is FALSE.
3rd: Use that as your given, and start proving
4th: When you come to a contradiction you have
proved that is true.
Example 2: Prove that If a > 0
then 1/a > 0
Statement
Example 1: Prove that a triangle
cannot have 2 right <‘s
Statement
1. Assume a
triangle has 2
right <‘s
2.m<1= m<2=90
3.m<1+m<2=180
4.m<1+m<2+m<
3 = 180
5. m<3= 0
6.----------------
Reason
1. Given
2. Definition right <
3. Substitution
4. Triangle sum
theorem
5. Contradiction
6. So a triangle
cannot have 2
right angles.
1. Assume a>0
so 1/a>0
2. 1/a > 0
∙a ∙a
3. 1<0
Reason
1. Given
2. Multiplicative
prop.
3. Contradiction,
so a>0 then
1/a>0 is false.
Example 3
Prove that
BDA
is not a straight angle
Statement
1.< BDA is a
straight angle
2.m<BDA=180
3. AD is
perpendicular
to BC
4. <BDA is a
right angle
5. m<BDA= 90
6. m<BDA is not
a straight angle
7. BDA is not a
straight angle
Reason
1. Assume opposite
2. Definition of a
straight angle
3. Given
4. Def. of
perpendicular lines
5. Def. of a right <
6. Contradiction
7. We know by the
contradiction that
its not a straight
angle
-The
hinge theorem states that If two triangles have
two sides that are congruent, but the third side is
not congruent, then the triangle with the larger
included angle has the longer third side.
-Converse: If the triangle with larger included angle
has the longer third side, then the two triangles
have two sides that are congruent but the third side
is not congruent.
What can you prove by the hinge theorem?
Example 1
A
B
X
C Y
-You can prove that these triangles are the
same because the angles are equal/
B
Example 2
37°
A
Z
42°
Z
Example 3
J
G
50° 62°
H
Complete using <, > or =
JG < Ji
K
I
L
W
Which side has a greater measurement?
Line KW is longer because the angle is
bigger.
In 30-60-90 triangles the hypotenous
is 2 times of the shorter led, and the
length of the longer leg is the length
of the shorter leg times √3
-In 45-45-90 triangles both legs are
congruent , 3 the length of the
hypotenoues is the lengt of a leg
times √2.
-
Example 1
16
√2
16√2
2
=8√2
16
X
Example 3
X= 9√2
2
X
9
Example 2
X
X
3
3
3² + 3²=x²
9 + 9 =x²
√18 = √x²
x= 3 √2
Example 1
60°
22
x
Example 2
X= 11
Y= 11√3
X
18√3
30°
y
Y
Example 3
10√3
X= 10
Y= 20
X
Y
X=9√3
Y= 27