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Krittika Poksawat N0.2 Class M.6/4 Tatcha Tratornpisuttikul N0.12 Class M.6/4 Worawoot Sumontra N0.23 Class M.6/4

Mahidol Wittayanusorn

Constructing the quadrilateral with the

maximum area when given the lengths

Adviser

Miss Nongluck Arpasut Mr. Sunya Phumkumarn

Constructing the quadrilateral with the maximum area when given the lengths

Introduction

Currently, to find the area of any geometric figure is usually from any ready made figure. In another way, if the sides of figure are given then many geometric figures can be made. Our group will study how to construct maximal area figures especially in quadrilateral. The study is to examine that how the given four sides can be arranged and how to adjust the angles to construct a maximal area quadrilateral.


Objective

1. To study the arrangement of sides of the quadrilateral with the maximum area.

2. To study the relation of angles and sides of the quadrilateral with the maximum area.

3. To use the Mathematics knowledge to solve the problem.


)](2

1[cos))()()(( 2 BAabcddscsbsas

2

dcbas

Consider , the area of the quadrilateral

Area =

When given

Method

That is we must construct the quadrilateral in the circle.


))()()(( dscsbsas

2

dcbas

So that, we can find the maximum area.

Area =

When given

Method


Consider, the quadrilateral with the sum of the opposite angle equal to 180.

ab

c

d

m

n

o

p

Given the quadrilateral mnop with the sides, a,b,c,d .

Method


ab

c

d

p

o

n

m

ab

cdp

o

nm

Figure 1 Figure 2

When figure 1 change to figure 2, we can suppose that the sum of the opposite angle must equal to

Method


))()()(( dscsbsas

Method

1.) Consider the order of four sides. We found that it can be 3! or 6 figures and the sum of length of three sides of quadrilateral must more than another one. Thus every quadrilaterals can construct 6 figures.

2.) Consider the order of four sides. We found that the 6 figures must have the same maximum area . The maximum area can calculate from the formula that is


consider triangle ADC จาก law of cosine

then

consider triangle ABC from law of cosinethen

…..1

…..2(1) = (2)

d

a

b

c

X

C

A

B

D

Finding the relation

Bcddcx cos2222

Dabbax cos2222

cos2222 cddcx

)cos(2222 abbax

cos2222 abbax

cos2cos2 2222 abbacddc

)()()(cos2 2222 badccdab

)(2

)()(cos

2222

cdab

badc

)(2

)()(arccos

2222

cdab

badc

Method


Errrrx cos2222

)2cos(2222 rrrrx

)1cos2(22 2222 rrx

22222 2cos42 rrrx

)cos1(4 222 rx

222 sin4rx

consider triangle AEC from law of cosinethen

….(3)

thus

From 1=3

Finding radius of circle

d

a

b

c

r

rX

E

C

A

B

D

2

2222 sin4cos2 rcddc

sin2

cos2222

cddcr

Method


finding , , andgi

ving

is the angle between radius of circle in triangle CED

is the angle between radius of circle in triangle

AED

is the angle between radius of circle in triangle AEB

is the angle between radius of circle in triangle BEC

cos2222 rrrra

)cos1(2 22 ra

2

2

21cos

r

a

2

2

21arccos

r

a

Consider triangle CED of law of cosine then

thus

2

2

21arccos

r

b

2

2

21arccos

r

c

2

2

21arccos

r

d

In the same way

Finding the angle at the

center of circle

d

a

b

c

r

r

r

r

E

C

A

B

D

Method


2

cos( ) cos2

cos( ) cos( ) 2cos2

consider

then….(1)

(1)+(2) thus

….(2)

2 2

( ) ( ) ( ) ( )2cos cos 2 os2c

2

2 2

( ) ( )cos cos os2c

2

( ) ( )cos os2c

2

( ) ( )cos os2c

2

( ) ( ) 22

( ) ( ) 2 4

Finding the relation of angles

d

a

b

c

r

r

r

r

E

C

A

B

D

22

)22cos()cos(

)22cos(2cos)cos()cos(

Method


d

a

b

c

C

A

B

D

)(2

)()(arccos

2222

cdab

badc

The relationship between angles and sides of the quadrilateral in the circle when the angle is between two sides that are adjacent sides.

The first result


The relationship among the radius of the circle, the angle and the length of the quadrilateral is

r

d

a

c

r

b

C

E

B

D

A

The second result

sin2

cos2222

cddcr


The relationship between the angle at the center of circle and the length of quadrilateral and the radius of circle is

2

2arccos 1

2

a

r

2

2arccos 1

2

b

r

2

2arccos 1

2

c

r

2

2arccos 1

2

d

r

r

d

a

c

r

b

r

r

C

E

B

D

A

The third result


When we know the four lengths of quadrilateral, we can construct the quadrilateral which has the maximum areas by construct it in the circle and how to construct is here.

The forth result


1. Construct the circle and the radius of the circle, calculated from the formula above.

r

E

D

The fourth result


2 . Draw two radius of the circle and the angle between them, calculated from the formula above, then draw line connect the end of two radius, so we got the first side of the quadrilateral.

The fourth result

r

a

r

C

E

D

rr

C

E

D


3. In the same way : draw the another radius of circle, one radius per time and the angle between two radius are , and , so we got the quadrilateral which has the maximum area.

The fourth result

r

a

r

r

C

E

D

A

r

a

r

br

C

E

D

A

r

a

r

b

r

r

C

E

B

D

A

r

a

c

r

b

r

r

C

E

B

D

A

r

d

a

c

r

b

r

r

C

E

B

D

A


The relationship between the angle at the center of circle and the angle in the quadrilateral is

( ) ( ) 2 4

r

d

a

c

r

b

r

r

C

E

B

D

A

The fifth result


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Entertainment & Humor

lengths of quadrilateral

maximum areawhen

quadrilateral mnopwith

maximal area quadrilateral

maximum areas

radius of circle method

center of circle method

sides of figure