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: dszhao@nie.edu.sg