# Chapter1: Triangle Midpoint Theorem and Intercept Theorem - PowerPoint PPT Presentation

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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

13
• Given in Figure ?BAF?DAF.
• 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,
• 2. ?AFC?ABC?BAF (Exterior angle )
• 2?BAF ?ABC (by 1)
• 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