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Q: What is the size of each individual interior angle of a polygon having 464 diagonals showing key stages of work?

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Suppose the polygon has V vertices.Then sum of interior angles is (V - 2)*180 degrees = 1980 degrees => V - 2 = 1980/180 = 11 => V = 13 A polygon with V vertices has V*(V-3)/2 = 13*10/2 = 65 diagonals.

It must have 15 sides to comply with the formula:- 0.5*(152-(3*15)) = 90 diagonals So: 360/15 = 24 degrees and 180-24 = 156 degrees Therefore: each interior angle measures 156 degrees

1 Let the sides be n and use the formula: 0.5*(n2-3n) = diagonals 2 So: 0.5*(n2-3n) = 170 => which transposes to: n2-3n-340 = 0 3 Solving the above quadratic equation gives n a positive value of 20 4 So the polygon has 20 sides and (20-2)*180 = 3240 interior angles 5 Each interior angle measures: 3240/20 = 162 degrees

It is: (5040+360)/180 = 30 sides

Let its sides be x and use the formula: 0.5*(x squared-3x) = 230 So: x squared-3x-460 = 0 Solving the quadratic equation gives x positive value of 23 Therefore the polygon has 23 sides irrespective of it being a regular or an irregular polygon. Check: 0.5*(23^2-(3*23)) = 230 diagonals

Let the number of sides be n and so:- If: 0.5*(n^2 -3n) = 275 Then: n^2 -3n -550 = 0 Solving the above quadratic equation: n has positive value of 25 Each interior angle: (25-2)*180/25 = 165.6 degrees

Using the diagonal formula when n is number of sides :- If: 0.5*(n^2-3n) = 189 Then multiplying both sides by 2 and subtracting both sides by 2*189 So: n^2-3n-378 = 0 Solving the above quadratic equation gives n a positive value of 21 Sum of interior angles: (21-2)*180 = 3420 degrees

2520 to get this. You must get the number of sides(16) subtract it by 2, then multiply it by 180. Subtracting it by two is actually the number of triangles inside the polygon showing the possible number of all available non-intersecting diagonals. so in general. The number of non-intersecting triangles multiplied by 180 degrees.( which is the number of degrees in one triangle.

Let its sides be x and rearrange the diagonal formula into a quadratic equation:- So: 0.5(x^2-3x) = 252 Then: x^2-3n-504 = 0 Solving the quadratic equation: gives x a positive value of 24 Therefore the polygon has 24 sides irrespective of it being irregular or regular

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Let the number of sides be x and use the diagonal formula:- If: 0.5*(x^2 -3x) = 90 then x^2 -3x = 90*2 If: x^2 -3x = 180 then x^2 -3x -180 = 0 Using the quadratic equation formula gives a positive value of 15 for x Therefore perimeter of the polygon: 15*4 = 60 cm

Let the diagonals be x+5 and x:- If: 0.5*(x+5)*x = 150 sq cm Then: x2+5x-300 = 0 Solving the above by means of the quadratic equation formula: x = +15 Therefore: diagonals are 15 cm and 20 cm The rhombus has 4 interior right angle triangles each having an hypotenuse Dimensions of their sides: 7.5 and 10 cm Using Pythagoras' theorem: 7.52+102 = 156.25 Its square root: 12.5 cm Thus: 4*12.5 = 50 cm which is the perimeter of the rhombus Note: area of any quadrilateral whose diagonals are perpendicular is 0.5*product of their diagonals

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