Chapter1: Triangle Midpoint Theorem and Intercept Theorem

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Chapter1 Triangle Midpoint Theorem and
Intercept Theorem

• Outline
• Basic concepts and facts
• Proof 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 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

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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 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

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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.

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

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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
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• Example 2 In the triangle ABC, D is an
  interior point of BC. AF bisects ?BAD. Show
  that ?ABC?ADC2?AFC.

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• Given in Figure ?BAF?DAF.
• To prove ?ABC?ADC2?AFC.
• 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 ?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)

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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 derived from the previous
  statements ,
• or an accepted statement.
• The last statement in the sequence is the
• conclusion.

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1.3. Midpoint Theorem
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 triangle
• is parallel to the third side
• and
• is half as long as the third side.

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

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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.
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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.

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• 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
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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
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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(?) .

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

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• (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

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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 AFa and FDb.

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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 [phone] E-mail
dszhao_at_nie.edu.sg