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Geometry: Special Segments and Centers in Triangles, Study notes of Geometry

University of the Philippines Visayas - Cebu High School (UPVC)Geometry

Definitions and properties of various segments and centers in triangles, including perpendicular bisectors, angle bisectors, medians, altitudes, and points of concurrency. It covers the classifications of triangles and explains the relationships between the circumcenter, incenter, and orthocenter.

What you will learn

  • What is a perpendicular bisector in a triangle?
  • What are the properties of the circumcenter, incenter, and orthocenter in a triangle?
  • What is the point of concurrency of the angle bisectors called?

Typology: Study notes

2021/2022

Uploaded on 08/05/2022

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hal_s95 🇵🇭

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bg1
1
Jul249:36AM
3.1SpecialSegmentsandCentersof
Triangles
ICAN...
Defineandrecognizeperpendicular
bisectors,anglebisectors,medians,
andaltitudes.
Defineandrecognizepointsof
concurrency.
Jul249:36AM
ClassificationsofTriangles:
BySide:
1.
Equilateral
:Atrianglewiththree
congruentsides.
2.
Isosceles
:Atrianglewithatleast
twocongruentsides.
3.
Scalene
:Atrianglewiththreesides
havingdifferentlengths.(nosidesare
congruent)
Jul249:36AM
ClassificationsofTriangles:
Byangle
1.
Acute
:Atrianglewiththreeacute
angles.
2.
Obtuse
:Atrianglewithoneobtuse
angle.
3.
Right
:Atrianglewithonerightangle
4.
Equiangular
:Atrianglewiththree
congruentangles
Jul249:36AM
SpecialSegmentsandCentersin
Triangles
A
PerpendicularBisector
isasegmentor
line
thatpassesthroughthemidpointofasideand
isperpendiculartothatside.
Jul249:36AM
Twolinesintersectatapoint.
When
three
ormorelinesintersectatthe
samepoint,itiscalleda
"Pointof
Concurrency."
PointofConcurrency
Jul249:36AM

Thethree
perpendicularbisectors
of
atriangleintersectatasinglepoint.
Thepointof
concurrencyofthe
perpendicular
bisectorsiscalled
the
circumcenter
.
pf3

Partial preview of the text

Download Geometry: Special Segments and Centers in Triangles and more Study notes Geometry in PDF only on Docsity!

Jul 249:36 AM

3.1 Special Segments and Centers of

Triangles

I CAN...

Define and recognize perpendicular

bisectors, angle bisectors, medians,

and altitudes.

Define and recognize points of

concurrency.

Jul 249:36 AM

Classifications of Triangles:

By Side:

  1. Equilateral : A triangle with three

congruent sides.

  1. Isosceles: A triangle with at least

two congruent sides.

  1. Scalene: A triangle with three sides

having different lengths. (no sides are

congruent)

Jul 249:36 AM

Classifications of Triangles:

By angle

  1. Acute: A triangle with three acute

angles.

  1. Obtuse: A triangle with one obtuse

angle.

  1. Right: A triangle with one right angle
  2. Equiangular: A triangle with three

congruent angles

Jul 249:36 AM

Special Segments and Centers in Triangles

A Perpendicular Bisector is a segment or line

that passes through the midpoint of a side and

is perpendicular to that side.

Jul 249:36 AM

Two lines intersect at a point.

When three or more lines intersect at the

same point, it is called a "Point of

Concurrency."

Point of Concurrency

Jul 249:36 AM

The three perpendicular bisectors of

a triangle intersect at a single point.

The point of

concurrency of the

perpendicular

bisectors is called

the circumcenter.

Jul 249:36 AM

Circumcenter Properties

  1. The circumcenter is

the center of the

circumscribed circle.

  1. The circumcenter is

equidistant to each of

the triangles vertices.

Jul 249:36 AM

An angle bisector is a segment that divides

an angle into two congruent angles.

m ∠ABD= m ∠DBC

BD is an angle bisector.

Jul 249:36 AM

The three angle bisectors of a triangle

intersect at a single point.

The point of concurrency of the angle

bisectors is called the incenter.

Point A is the incenter of the triangle

Jul 249:36 AM

Incenter properties

  1. The incenter is the

center of the

inscribed circle

  1. The incenter is

equidistant to each

side of the triangle.

AB = AD = AC

Jul 249:36 AM

An altitude is a segment from a

vertex perpendicular to the opposite side

AD is an altitude of ∆ABC

m ∠ADB= m ∠ADC=90°

Jul 249:36 AM

The three altitudes of a triangle are

concurrent. The point of concurrency is

called the orthocenter.

Point A is the orthocenter

of the triangle