Definition of Perpendicular bisector: a line perpendicular to a segment at the line segment´s...
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Transcript of Definition of Perpendicular bisector: a line perpendicular to a segment at the line segment´s...
Chapter 5 Journal
By: Ana Cristina Andrade
Perpendicular Bisector:Definition of Perpendicular bisector: a line perpendicular to a segment at the line segment´s midpoint
Perpendicular bisector theorem: If a line is perpendicular, then it is equidistant from the endpoints of a segment.
Converse of perpendicular bisector theorem: If a point is equidistant from the endpoint of a segment, then it is perpendicular line.
Examples:
1. =2. =3. =
Perpendicular bisector theorem
Examples:Converse of perpendicular
bisector theorem
Angle Bisector:Angle bisector theorem: a ray or line that cuts an angle into 2 congruent angles. It always lies on the inside of an angle
Converse of angle bisector theorem: If a point is equidistant from the sides of a angle, then it lies on the bisector angle.
Examples: Angle bisector theorem
AB
C
A
B
CAB
C
AB = CB
Examples:Converse of
angle bisector theorem
A
B
C
D
<ADB = <CDB(Congruent ,not equal)
A
B
C
D
A
B
C
D
Concurrency:Definition of concurrency: Where three or more lines intersect at one point.
concurrency of Perpendicular bisectors:Concurrency of perpendicular bisectors: Point where the perpendicular bisectors intersect.
Circumcenter:Definition of Circumcenter: the point of congruency where the perpendicular bisectors of a triangle meet.The circumcenter theorem: The circumcenter of a triangle is equidistant from the vertices of the triangle.
Examples:
concurrency of angle bisectors:Concurrency of angle bisectors: Point where the angle bisectors intersect.
Incenter:Definition of incenter: The point where the angle bisector intersect of a triangleAlways occur on the side of triangleIncenter theorem: The incenter of a triangle is equidistant from the side of a triangle
Examples:
Median:Definition of Median: segment that goes from the vertex of a triangle to the opposite midpoint.
Centroid:Centroid: The point where the medians of a triangle intersect.The distance from the vertex to the centroid is double the distance from the centroid to the opposite midpoint.
Examples:
concurrency of medians:Concurrency of medians: point where the medians intersect.
Altitude:
Definition of altitude: a segment that goes from the vertex perpendicular to the line containing the opposite side.
Examples:
Orthocenter:Definition of Orthocenter: Where the altitudes intersectIf the triangle is acute, the orthocenter is on the inside of the triangleIf it is right orthocenter is on the vertex of the right angle.
Examples:
concurrency of altitudes:Concurrency of altitudes: point where the altitudes intersect.
Midsegment:Midsegment of a triangle: segment that joins the midpoints of two sides of the triangleA midsegment of a triangle, and its length is half the length of that side.
Examples:
midsegment theorem:Triangle midsegment theorem: A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.
relationship between the longer and shorter sides of a triangle:
Hinge theorem: If 2 triangles have 2 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 of Hinge theorem: If two sides of a triangle are congruent to the two sides of the other triangle but the other sides are not congruent then, the largest included angle is across from the largest side.
Examples:
<B > <Y
AC > XZ
A
B
C
H
I
J
AB
C
H
I J
J>A
HI>B
C
Hinge Theorem
Examples: Converse of Hinge Theorem
A
B
C
D
E
F
FE > CB, FD=CA, DE = AB (congruent)
<D> <A
A
BC
D
F E
B
ED
AC
FE
Relationship between opposite angles of a triangle:
Triangle side-angle relationship theorem: In any triangle, the longest side is always opposite from the largest angle and vice versa.
Examples:
Longest side
Shortest side
Longest side
exterior angle inequality:The non-adjacent interior angles are smaller than the exterior angleA+B = exterior angle (c)
A B CA
CB
A
BC
Triangle inequality:
Triangle inequality theorem: the 2 smaller sides of a triangle must add up to more than the length of the 3rd side.
Examples:4, 7, 104+7=11
YES2, 9, 122+9=11
NO
3, 1.1, 1.71.1+1.7= 2.8
NO
indirect proof:Indirect proof: used when it is not possible to prove something directly.
Steps:1.Assume that what you are proving is false2.Use that as your given, and start proving it3.When you come to a contradiction you have
proved that it is true.
Examples:Prove: A triangle cannot have 2 right angles
A triangle has 2 right angles (<1 & <2)
Given
M<1=m<2=90 Def. right angle
M<1+m<2=180 Substitution
M<1+m<2+m<3=180 Triangle sum theorem
M<3=0 contradiction
Examples:Proove: a right triangle cannot have an obtuse angle
A right triangle can have an obtuse angle (<A)
Given
M<A + m<B= 90 Substitution
M<A =90 – m<b Subtraction prop.
M<A> 90° Def. obtuse triangle
90° - m<b > 90 substitution
m<b = 0 contradiction
Examples:A triangle cannot have 4 sides
A triangle can have 4 sides
Given
A square is a shape with 4 sides
Def of square
A triangle is a shape with only 3 sides
Def of triangle
A triangle cannot have 4 sides
contradiction
special relationships in the special right triangles:
45° - 45° - 90° triangle theorem: In this kind of triangle, both legs are congruent and the hypotenuse is the length of a leg times √230° - 60° - 90° Triangle theorem: In this kind of triangle the longest leg is √3 the shorter leg and the hypotenuse is √2 the shortest side of the triangle.
Examples:
X
X
BC=AC=XAB=X√2
A
B C
45° - 45° - 90° triangle theorem
Examples:
45°
45°
14X
X=14√2
45° - 45° - 90° triangle theorem
Examples:30° - 60° - 90° triangle theorem
B16
16=2a8=aB=a√3B=8√3
Y
2020=2x10=xY=a√3Y=10√3
d
100
100=2d50=dH=d√3H=50√3