British Flag Theorem [PDF]

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BRITISH FLAG THEOREM Given any point P that lies inside rectangle ABCD, we have: AP²+CP²=BP²+DP². The measurements AP, BP, CP, and DP denote the distances of P from each of the rectangle's vertices. Exercises: 1. Point E lies within rectangle ABCD. If AE=7, BE=5, and CE=8, what is DE?



2. Let P be a point inside rectangle ABCD. If PD=12, PC=7 and PA=10, what is PB?



3. O is any point inside rectangle ABCD such that OB =6cm, OD = 8cm and OA =5cm. What will be the length OC?



4. Inside of the rectangle IMTJ, we select a point G such that GI=7, GM=15, GT=24 then GJ=?



5. Inside Rectangle CEDI, a point L exists such that CL=6 cm, EL=4 cm, and DL=5 cm. What is the length of IL?



6. Let P be a point inside rectangle ABCD. If AP=10, CP=7, and DP=12 cm, find BP.



7. In rectangle CORE, a point N is chosen inside such that CN=RN=5 in and EN=7 in. Find the length of ON, in inches.



8. In rectangle ABCD, a point P is chosen such that AP=2 cm, BP=3 cm, and CP=4 cm. Find the length of DP.



9. A rectangle MNOP has a point K insides such that the length of MK=x+1, NK=x+2, OK=x+3, and PK=x+4. Find the length of PK.



VIVIANI'S THEOREM This theorem states that for any point P inside an EQUILATERAL triangle ABC, the sum of the distances of P from AB, BC, and AC is equal to its height.



Exercises: 1. A point P is chosen from an equilateral triangle such that its distances from each side are 5 cm, 8 cm, and 11 cm. Find the length of its altitude.



2. The height of an equilateral triangle is 14√3 cm. If a point has distance 3√3 and 5√3 cm from two of its sides, find the distance of P from the third side.



3. A tree is enclosed within the perimeter of posts A, B, and C, each equidistant from one another and wired by a fence. The tree is 20 meters from the fence connected by A and B, 30 meters from the fence connected by B and C, and 40 meters from the fence connected by A and C.



4. The distance of a point from an equilateral triangle with height 9 cm from two of its sides are equal, and the distance of the same point from the third side is 7 cm. Find the common distance from the point.



5. Find the area of an equilateral triangle such that there exists a point whose distance from each sides are 6 cm, 8 cm, and 11 cm.



Trigonometric equations http://www.sosmath.com/algebra/solve/solve0/solvtrig.html



http://tutorial.math.lamar.edu/Problems/CalcI/TrigEquations.aspx



Quadratic Equations http://tutorial.math.lamar.edu/Problems/Alg/SolveQuadraticEqnsI.aspx



Geometry Problems https://www.analyzemath.com/high_school_math/grade_10/geometry .html



Geometry Word Problems https://www.purplemath.com/modules/perimetr2.htm