The document discusses constructing a quadrilateral with the maximum area given the lengths of its four sides. It presents the objective of studying how to arrange the sides and adjust the angles. The method described is to construct the quadrilateral within a circle since the area is maximized when the opposite angles sum to 180 degrees. Formulas are derived relating the radius, angles at the center and between radii, and side lengths of the inscribed quadrilateral with maximum area.

1. 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

2. Adviser Miss Nongluk Arpasuth Mr. Sunya Phumkumarn Constructing the quadrilateral with the maximum area when given the lengths

3. 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. Constructing the quadrilateral with the maximum area when given the lengths

4. Objective To study the arrangement of sides of the quadrilateral with the maximum area. 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. Constructing the quadrilateral with the maximum area when given the lengths

5. ระยะเวลาในการดำเนินงาน From July 2007 to December 2007 Constructing the quadrilateral with the maximum area when given the lengths

6. that cos ( A+B) equal to 0 . So that Consider , the area of the quadrilateral Area = When given The area will be the maximum when Constructing the quadrilateral with the maximum area when given the lengths Method is the smallest That is we must construct the quadrilateral in the circle.

7. So that, we can find the maximum area Area = When given Constructing the quadrilateral with the maximum area when given the lengths Method

8. Consider, the quadrilateral with the sum of the opposite angle equal to 180 Given the quadrilateral mnop with the sides, a,b,c,d . Constructing the quadrilateral with the maximum area when given the lengths Method

9. Figure 1 Figure 2 consider figure 1 change to figure 2 we can suppose that the sum of the opposite angle must equal to Constructing the quadrilateral with the maximum area when given the lengths Method

10. Constructing the quadrilateral with the maximum area when given the lengths Method 1.)Consider the order of four sides we can calculate that the most of figure can be 3!or 6. And we found 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 figure must have the same maximum area .Consider from the formula of maximum area when

11. consider triangle ADC จาก law of cosine then consider triangle ABC from law of cosine then … ..1 … ..2 (1) = (2) when Step finding the relation Constructing the quadrilateral with the maximum area when given the lengths Method

12. consider triangle AEC from law of cosine then … .(3) thus From 1=3 Finding radius of circle Constructing the quadrilateral with the maximum area when given the lengths Method

13. finding , , and giving is the angle between radii of circle in triangle CED is the angle between radii of circle in triangle AED is the angle between radii of circle in triangle AEB is the angle between radii of circle in triangle BEC Consider triangle CED of law of cosine then thus In the same way Finding the angle at the center of circle Constructing the quadrilateral with the maximum area when given the lengths Method

14. consider then … .(1) (1)+(2) thus … .(2) Finding the relation of angles Constructing the quadrilateral with the maximum area when given the lengths Method

15. when ผลการศึกษาที่ 1 Constructing the quadrilateral with the maximum area when given the lengths Relation between angles and the sides of quadrilateral in the circle when the angle is between two sides that are adjacent sides

16. The relation among the radius of the circle and the angle and the length of the quadrilateral is When r is the radius of the circle c,d is the length that close to the angle given The second result Constructing the quadrilateral with the maximum area when given the lengths

17. The relation among the angle at the center of circle and the length of quadrilateral and the radius of circle is ผลการศึกษาที่ 3 Constructing the quadrilateral with the maximum area when given the lengths

18. When we know the four lengths of quadrilateral and want to construct the quadrilateral which has the maximum areas, we have to construct it in the circle and how to construct it is ผลการศึกษาที่ 4 Constructing the quadrilateral with the maximum area when given the lengths

19. 1. Construct the circle and the radius of the circle can calculate from the formula above The fourth result Constructing the quadrilateral with the maximum area when given the lengths

20. 2 . Draw two radius of the circle and the angle between them can calculate 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 Constructing the quadrilateral with the maximum area when given the lengths

21. 3. In the same way of item two : drawing the radius of circle increasing 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 Constructing the quadrilateral with the maximum area when given the lengths

22. The relation between the angle at the center of circle and the angle in the quadrilateral is The fifth result Constructing the quadrilateral with the maximum area when given the lengths