presentation triangle midpoint theorem
TRANSCRIPT
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Chapter1: Triangle Midpoint Theorem
and Intercept Theorem
Outline
Basic concepts and factsProof and presentation
Midpoint Theorem
Intercept Theorem
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1.1. Basic concepts and facts
In-Class-Activity 1.
(a) State the definition of the following terms:
Parallel lines,
Congruent triangles,
Similar triangles:
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Two lines are parallelif they do not meet
at any point
Two triangles are congruentif their
corresponding angles and correspondingsides equal
Two triangles are similarif their
Corresponding angles equal and theircorresponding sides are in proportion.
[Figure1]
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(b) List as many sufficient conditions as
possible for
two lines to be parallel,
two triangles to be congruent,
two triangles to be similar
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Conditions for lines two be parallel
two lines perpendicular to the same line.
two lines parallel to a third line
If two lines are cut by a t ransversal ,
(a) two alternat ive in ter ior (exter ior) ang lesare
equal.
(b) two corresponding anglesare equal
(c) two in ter ior angles on the same side of
the transversal are supplement
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Corresponding angles
Alternative angles
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Conditions for two triangles to be congruent
S.A.S
A.S.A
S.S.S
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Conditions for two triangles similar
Similar to the same triangle
A.A
S.A.S
S.S.S
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1.2. Proofs and presentation
What is a proof? How to present a proof?
Example 1 Suppose in the figure ,
CD is a bisector of and CD
is perpendicular to AB. Prove AC is equal
to CB.
ACB
DA B
C
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Given the figure in which
To prove that AC=BC.
The plan is to prove that
ABCDBCDACD ,
BCDACD
DA B
C
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Proof
1.
2.
3.4.
5. CD=CD
6.7. AC=BC
1. Given
2. Given
3. By 24. By 2
5. Same segment
6. A.S.A7. Corresponding sides
of congruent
triangles are equal
BCDACD
ABCD
090CDA0
90CDB
BCDACD
Statements ReasonsDA B
C
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Example 2 In the triangle ABC, D is an
interior point of BC. AF bisects
BAD.Show that ABC+ADC=2AFC.
A C
B
D
F
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Given in Figure BAF=DAF.
To prove ABC+ADC=2AFC.
The plan is to use the properties of angles in
a triangle
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Proof: (Another format of presenting a proof)
1. AF is a bisector of BAD,
so BAD=2BAF.2. AFC=ABC+BAF (Exterior angle)
3. ADC=BAD+ABC (Exterior angle)
=2BAF +ABC (by 1)
4. ADC+ABC
=2BAF +ABC+ ABC ( by 3)
=2BAF +2ABC=2(BAF +ABC)
=2AFC. (by 2)
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What is a proof?
A proof is a sequence of statements, where
each statement is either
an assumption,
or a statement derivedfrom the previousstatements ,
or an accepted statement.
The last statement in the sequence is theconclusion.
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1.3. Midpoint Theorem
ED
A B
C
Figure2
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1.3. Midpoint Theorem
Theorem 1[ Triangle Midpoint Theorem]
The line segment connecting the midpoints
of two sides of a triangleis parallel to the third side
and
is half as long as the third side.
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Given in the figure , AD=CD, BE=CE.
To prove DE// AB and DE=
Plan: to prove ~
AB2
1
ACB DCE
ED
A B
C
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Proof
Statements Reasons
1.
2. AC:DC=BC:EC=2
4. ~
5.
6. DE // AB
7. DE:AB=DC:CA=2
8. DE= 1/2AB
1. Same angle
2. Given
4. S.A.S
5. Corresponding
angles of similar
triangles
6. corresponding angles7. By 4 and 2
8. By 7.
DCEACB
ACB DCE
CDECAB
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Example3 The median of a trapezoid is
parallel to the bases and equal to one halfof the sum of bases.
FE
CD
A B
Complete the proof
Figure
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Example 4 ( Right triangle median theorem)
The measure of the median on thehypotenuse of a right triangle is one-half of
the measure of the hypotenuse.
E
A
C
B
Read the proof on the notes
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In-Class-Activity 4
(posing the converse problem)
Suppose in a triangle the measure of a
median on a side is one-half of the measure
of that side. Is the triangle a right
triangle?
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1.4 Triangle Intercept Theorem
Theorem 2[Triangle Intercept Theorem]
If a line is parallel to one side of a triangle
it divides the other two sides proportionally.Also converse(?) .
B
C
D E
A
Figure
Write down the complete
proof
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Example 5 IntriangleABC, suppose
AE=BF, AC//EK//FJ.(a) Prove CK=BJ.
(b) Prove EK+FJ=AC.
J
K
A
C
BE F
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(a)
1
2.
3.
4.5.
6.
7. Ck=BJ(b) Link the mid points of EF and KJ. Then use
the midline theorem for trapezoid
BF
EF
BJ
KJ
BF
BE
BJ
BK
BK
CK
BE
AE
BK
BE
CK
AE
BJ
BF
CK
AE
1BF
AE
BJ
CK
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In-Class-Exercise
In , the points D and F are on side AB,
point E is on side AC.
(1) Suppose that
Draw the figure, then find DB.
( 2 ) Find DB if AF=a and FD=b.
ABC
6,4,//,// FDAFDCFEBCDE
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Please submit the solutions of
(1) In class-exercise on pg 7(2) another 4 problems in
Tutorial 1
next time.
THANK YOU
Zhao Dongsheng
MME/NIE
Tel: 67903893
E-mail: [email protected]