The document discusses constructing a quadrilateral with the maximum area given the lengths of its four sides. It presents the following key points: 1) The maximum area quadrilateral can be constructed by arranging the sides in a circle, with the opposite angles summing to 180 degrees. 2) The relationships between the radius of the circle, the angles between sides, and the side lengths are described by formulas using trigonometry. 3) Given the four side lengths, the maximum area quadrilateral can be constructed in a circle by first calculating the radius, then drawing the sides based on the radius and angle formulas.

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 Nongluck Arpasut 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

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 When 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 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 when

11. consider triangle ADC จาก law of cosine then consider triangle ABC from law of cosine then … ..1 … ..2 (1) = (2) when 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 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 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 The relationship between angles and sides of the quadrilateral in the circle when the angle is between two sides that are adjacent sides.