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Chapter1: Triangle Midpoint Theorem and Intercept Theorem
Outline
•Basic concepts and facts
•Proof and presentation
•Midpoint Theorem
•Intercept Theorem
1.1. Basic concepts and facts
In-Class-Activity 1.
(a) State the definition of the following terms:
Parallel lines,
Congruent triangles,
Similar triangles:
•Two lines are parallel if they do not meet at any point
•Two triangles are congruent if their corresponding angles and corresponding sides equal
•Two triangles are similar if their
Corresponding angles equal and their corresponding sides are in proportion.
[Figure1]
(b) List as many sufficient conditions as possible for
• two lines to be parallel,
• two triangles to be congruent,
• two triangles to be similar
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 transversal ,
(a) two alternative interior (exterior) angles are
equal.
(b) two corresponding angles are equal
(c) two interior angles on the same side of
the transversal are supplement
Corresponding angles
Alternative angles
Conditions for two triangles to be congruent
• S.A.S
• A.S.A
• S.S.S
Conditions for two triangles similar
• Similar to the same triangle
• A.A
• S.A.S
• S.S.S
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
Given the figure in which
To prove that AC=BC.
The plan is to prove that
ABCDBCDACD ,
BCDACD
DA B
C
Proof
1.
2.
3.
4.
5. CD=CD
6.
7. AC=BC
1. Given
2. Given
3. By 2
4. By 2
5. Same segment
6. A.S.A
7. Corresponding sides
of congruent triangles are equal
BCDACD ABCD
090CDA090CDB
BCDACD
Statements Reasons DA B
C
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
Given in Figure BAF=DAF.
To prove ABC+ADC=2AFC.
The plan is to use the properties of angles in a triangle
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)
What is a proof?
A proof is a sequence of statements, where each statement is either
an assumption,
or a statement derived from the previous statements ,
or an accepted statement.
The last statement in the sequence is the
conclusion.
1.3. Midpoint Theorem
ED
A B
C
Figure2
1.3. Midpoint Theorem
Theorem 1 [ Triangle Midpoint Theorem]
The line segment connecting the midpoints
of two sides of a triangle
is parallel to the third side
and
is half as long as the third side.
Given in the figure , AD=CD, BE=CE.
To prove DE// AB and DE=
Plan: to prove ~
AB21
ACB DCE
ED
A B
C
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 angles
7. By 4 and 2
8. By 7.
DCEACB
ACB DCECDECAB
In-Class Activity 2 (Generalization and extension)
• If in the midpoint theorem we assume AD and BE are one quarter of AC and BC respectively, how should we change the conclusions?
• State and prove a general theorem of which the midpoint theorem is a special case.
Example 3 The median of a trapezoid is parallel to the bases and equal to one half of the sum of bases.
FE
CD
A B
Complete the proof
Figure
Example 4 ( Right triangle median theorem)
The measure of the median on the
hypotenuse of a right triangle is one-half of
the measure of the hypotenuse.
E
A
C
B
Read the proof on the notes
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?
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
Example 5 In triangle ABC, suppose AE=BF, AC//EK//FJ.
(a) Prove CK=BJ.
(b) Prove EK+FJ=AC.
J
K
A
C
BE F
(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
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
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]
