Centroid Of A Triangle Ppt

Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles - PowerPoint PPT Presentation

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Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles

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(O is on the perpendicular bisects of AB and BC) ( By 4 ) (Aforementioned segment ) ( M is the midpoint ) ... In-Grade-Exercise one. Prove Theorem ane ... In-Class-Exercise two ... – PowerPoint PPT presentation

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Title: Chapter 2 : Circumcenter, Orthocenter, incenter, and centroid of triangles

1
Chapter 2 Circumcenter, Orthocenter,
incenter, and centroid of triangles

Outline
Perpendicular bisector ,
circumcentre and orthocenter
Bisectors of angles and the incentre
Medians and centroid

ii
2.1 Perpendicular bisector, Circumcenter and
orthocenter of a triangle

Definition 1 The perpendicular bisector
of a line segment is a line perpendicular
to the line segment at its midpoint.

CD is a perpendicular bisector of AB if (i)
ACBC (2) DCA DCB
3
In-Class-Activity 1

(1) If P is a bespeak on the perpendicular
bisector of AB, what is the relationship
between PA and PB?
(ii) Brand a conjecture from the ascertainment in
(i). Prove the conjecture.
(3) What is the converse of the conjecture in
(two).
Can you lot bear witness it?

Theorem 1 The perpendicular bisectors of the
three sides of a triangle meet at a betoken
which is as distant from the vertices
of the triangle.

The point of intersection of the three
perpendicular bisectors of a triangle is called
the circumcenter of the triangle.
5

DG, MH and EF are the perpendicular
bisectors of the sides AB,AC and BC
respectively
DG, MH and EF meet at a betoken O
OAOBOC
O is the circumcenter of triangle ABC.

vi
Proof of Theorem i

Given in ABC that DG, EF and MH are the
perpendicular bisectors of sides AB, BC and CA
respectively.
To prove that
DG,EF and MH meet at a betoken O,
and AOBOCO.
Plan Allow DG and EF meet at a point O. So show
that OM is perpendicular to AC.

7
Proof

1.Permit DG and EF meet at O
2. Connect G and O.
We show MO is
perpendicular to side AC
3. Connect AO, BO and CO.

(If they dont run into, then DG//EF, and so AB//BC,
impossible)

4. AOBO, BOCO
5. AOCO
6. MOMO
7. AMCM
8.
nine.
10

(O is on the perpendicular bisects of AB and BC)
( By 4 )
(Same segment )
( M is the midpoint )
(Due south.S.S)
(Corresponding angles
(By 9 and )

nine

11.OM is the perpendicular (Two conditions
satisfied)
bisector of side Ac.
12. The iii perpendicular
bisector meet at indicate O.
xiii.O is as distant from ( by iv)
vertices A,B and C.

Remark 1 ( A method of proving that iii lines
meet at a indicate )
In order to prove three lines run into at one point,
we can
first name the meet betoken of two of the lines
then construct a line through the see point
(iii) terminal prove the constructed line coincides
with the third line.

In-Class-Exercise 1
Prove Theorem 1 for obtuse triangles.
Draw the effigy and requite the outline of the
proof

Remark two The circumcenter of a triangle is
as distant from the three vertices.
The circle whose centre is the
circumcenter of a triangle and whose radius is
the distance from the circumcenter to a vertex
is called the
circumscribed circle
of the triangle.

In-Class-Activity
Give the definition of parallelograms
(2) Listing equally many as possible conditions for a
quadrilateral to be a parallelogram.
(3) Listing any other backdrop of parallelogram
which are not listed in (2).

xiv
(1) Definition A parallelogram is a quadrilateral
with its opposite sides parallel
ABCD

(2) Conditions
The contrary sides equal
Opposite angles equal
The diagonals bisect each other
Two opposite side parallel and equal
(three)

Theorem ii
The three altitudes of a triangle meet at a
point.

sixteen

Given triangle ABC with altitudes Advertisement, Be
and CF.
To prove that AD, BE and CF meet at a point.
Program is to construct another larger triangle
ABC
such that Ad, Be and CF are the perpendicular
bisectors
of the sides of ABC. Then utilize
Theorem i.

Proof (Cursory)
Construct triangle ABC such that
AB//AB, Air-conditioning//Air conditioning, BC//BC
i. ABCB is a parallelogram.
2. BCAB.
3. Similarly CAAB.
iv. CE is the perpendicular bisector of ABC
of side BA.
5. Similarly BF and AD are perpendicular
bisectors of sides of ABC.
6. So AD, BF and CE meet at a signal (past
Theorem 1)

eighteen

The indicate of intersection of the three
altitudes of a triangle is called the
orthocenter
of the triangle.

xix
ii.2 Angle bisectors , the incenter of a
triangle

Angle bisector
ABD DBC
In-Class-Exercise ii
(1) Show that if P is a point on the bisector
of and so the distance from P to AB
equals the distance
from P to CB.
(two) Is the converse of the statement in (one)
also truthful?

Lemma 1 If AD and Exist are the bisectors of the
angles
A and B of ABC, then Advertisement and Exist
intersect at a point.

Proof Suppose they do non run into. i. A
B C180 ( Holding of triangles) ii.
Then Ad// BE. ( Definition of parallel
lines) 3. DAB EBA180 ( interior
angles on same side ) 4.
( AD and BE are bisectors
)

5.This contradicts that
The contradiction shows that the two angle
bisectors must see at a betoken.

Proof by contradiction ( Indirect proof) To
prove a argument by contradiction, we showtime
assume the statement is false, then deduce
two statements contradicting to each other.
Thus the original statement must exist truthful.
22

Theorem 3 The bisectors of the three angles of
a triangle
meet at a point that is every bit distant from the
iii side
of the triangle.

The point of intersection of angle bisectors of
a triangle is called the
incenter of the triangle
Read and complete the proof
23

Remark Suppose r is the distance from the
incenter to a side of a triangle. So there
is a circle whose center is the incenter and
whose radius is r.
This circumvolve tangents to the 3 sides
and is called the
inscribed circle ( or incircle) of the
triangle.

24
Example ane The sum of the altitude from any
interior signal of an equilateral triangle to the
sides of the triangle is abiding.
25

Proof
ane.
2.
3. ABACBC (ABC is equilateral )
4.
five.
( by 1 and 4)
6.
is a abiding.

In-Class-Activity
(1) State the converse of the conclusion proved
in Case 1.
Is the antipodal besides true?
Is the conclusion of Case ane true for points
outside the triangle?

27
two.3 Medians and centroid of a triangle

A median of a triangle is a line drawn from any
vertex to the mid-point of the opposite side.
Lemma 2 Whatsoever two medians of a triangle meet at a
point.

Theorem three The iii medians of a triangle see
at a point which is 2 tertiary of the distance
from each vertex to the mid-point of the opposite
side.

The point of intersection of the three medians of
a triangle is chosen the
centroid of the triangle
29

Proof (Outline)
Let 2 median AD and BE meet at O.
Prove
If CE and AE meet at O, then
So O is the same equally O
All medians laissez passer through O.
Read the proof

thirty

Example ii Allow line XYZ be parallel to side
BC and pass
through the centroid O of .
BX, AY and CZ are perpendicular to XYZ.
Prove AYBXCZ.

32
Question

Is the converse of the determination in
Case 2 likewise true?
How to evidence it?

Summary
The perpendicular bisectors of a triangle encounter at
a point---circumcenter, which is every bit afar
from the three vertices and is the center of the
circle outscribing the
triangle.
The iii altitudes of a triangle meet at a
bespeak--- orthocenter .
The angle bisectors of a triangle run across at a
point---incenter, which is equally afar from
the three sides and is the center of the circle
inscribed the triangle.
The three medians of a triangle meet at a point
---centroid. Physically, centroid is the heart
of mass of the triangle with uniform density.

34
Central terms

Perpendicular bisector
Bending bisector
Altitude
Median
Circumcenter
Orthocenter
Incenter
Centroid
Confining circle
Incircle

35
Delight submit the solutions of four
issues in Tutorial 2
next time. Give thanks YOU
Zhao Dongsheng MME/NIE Tel 67903893 Email
dszhao_at_nie.edu.sg

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