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Chapter1: Triangle Midpoint Theorem and Intercept Theorem

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Title: Chapter1: Triangle Midpoint Theorem and Intercept Theorem


1
Chapter1 Triangle Midpoint Theorem and
Intercept Theorem
  • Outline
  • Basic concepts and facts
  • Proof and presentation
  • Midpoint Theorem
  • Intercept Theorem

2
1.1. Basic concepts and facts
  • In-Class-Activity 1.
  • (a) State the definition of the following terms
  • Parallel lines,
  • Congruent triangles,
  • Similar triangles

3
  • 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

4
  • (b) List as many sufficient conditions as
    possible for
  • two lines to be parallel,
  • two triangles to be congruent,
  • two triangles to be similar

5
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

6
Corresponding angles
Alternative angles
7
Conditions for two triangles to be congruent
  • S.A.S
  • A.S.A
  • S.S.S

8
Conditions for two triangles similar
  • Similar to the same triangle
  • A.A
  • S.A.S
  • S.S.S

9
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.

10
  • Given the figure in which
  • To prove that ACBC.
  • The plan is to prove that

11
Proof
Statements
Reasons
1. 2. 3. 4. 5. CDCD 6. 7. ACBC 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
12
  • Example 2 In the triangle ABC, D is an
    interior point of BC. AF bisects ?BAD. Show
    that ?ABC?ADC2?AFC.

13
  • Given in Figure ?BAF?DAF.
  • To prove ?ABC?ADC2?AFC.
  • The plan is to use the properties of angles in a
    triangle

14
  • Proof (Another format of presenting a proof)
  • 1. AF is a bisector of ?BAD,
  • so ?BAD2?BAF.
  • 2. ?AFC?ABC?BAF (Exterior angle )
  • 3. ?ADC?BAD?ABC (Exterior angle)
  • 2?BAF ?ABC (by 1)
  • 4. ?ADC?ABC
  • 2?BAF ?ABC ?ABC ( by 3)
  • 2?BAF 2?ABC
  • 2(?BAF ?ABC)
  • 2?AFC.
    (by 2)

15
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.

16
1.3. Midpoint Theorem
Figure2
17
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.

18
  • Given in the figure , ADCD, BECE.
  • To prove DE// AB and DE
  • Plan to prove

19
Proof
Statements Reasons
1. 2. ACDCBCEC2 4. 5. 6. DE // AB 7. DEABDCCA2 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.
20
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.

21
  • Example 3 The median of a trapezoid is parallel
    to the bases and equal to one half of the sum of
    bases.

Figure
Complete the proof
22
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.

Read the proof on the notes
23
  • 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?

24
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(?) .

Figure
Write down the complete proof
25
  • Example 5 In triangle ABC, suppose AEBF,
    AC//EK//FJ.
  • (a) Prove CKBJ.
  • (b) Prove EKFJAC.

26
  • (a)
  • 1
  • 2.
  • 3.
  • 4.
  • 5.
  • 6.
  • 7. CkBJ
  • (b) Link the mid points of EF and KJ. Then use
  • the midline theorem for trapezoid

27
  • 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 AFa and FDb.

28
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
dszhao_at_nie.edu.sg
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