Presentation Triangle Midpoint Theorem

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 ACB and CD is perpendicular to AB. Prove AC is equal to CB. C

A

D

B

C

Given the figure in which ACD  BCD, CD  AB

To prove that AC=BC.

The plan

is to prove that

ACD  BCD

A

D

B

C

Proof Statements

A

1. ACD  BCD 2. CD  AB 3. CDA  900 4. CDB  900 5. CD=CD 6. ACD  BCD 7. AC=BC

D

B

Reasons

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

Example 2 In the triangle ABC, D is an interior point of BC. AF bisects BAD. Show that ABC+ADC=2AFC. B F D

A

C

Given in Figure BAF=DAF. To prove ABC+ADC=2AFC.

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=2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)

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 C

D

A

Figure2

E

B

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= 12 AB Plan: to prove ACB~ DCE C

D

A

E

B

Proof Statements 1. ACB  DCE 2. AC:DC=BC:EC=2 4. ACB ~ DCE 5. CAB  CDE

6. DE // AB 7. DE:AB=DC:CA=2 8. DE= 1/2AB

Reasons 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.

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. Figure A

E

D

Complete the proof

B

F

C

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. B

E

C A

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(?) . C

Figure D

Write down the complete proof

A

E

B

Example 5 In triangle ABC, suppose AE=BF, AC//EK//FJ. (a) Prove CK=BJ. (b) Prove EK+FJ=AC. C

K

J A

E

F

B

(a) 1 KJ  EF BJ BF 2. BK  BE BJ BF AE CK 3. BE  BK AE BE 4. CK  BK AE BF  5. CK BJ 6. 7. Ck=BJ CK AE  1 BJ BF

(b) Link the mid points of EF and KJ. Then use the midline theorem for trapezoid

In-Class-Exercise In ABC, the points D and F are on side AB, point E is on side AC. (1) Suppose that

DE // BC , FE // DC , AF  4, FD  6 Draw the figure, then find DB. ( 2 ) Find DB if AF=a

and FD=b.

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]