Prove if a circle is divided into n congruent arcs (n \ ge 3), the chords determined by joining consecutive end points of these arcs they form a regular polygon?

in Geometry Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

If the radius of the circle is r, its circumference is 2πr, so the length of each arc is 2πr/n. When the endpoints of the arc are joined to the centre of the circle they form radii. Each arc subtends an angle 2π/n at the centre of the circle. The chord joining the endpoints and the two radii form an isosceles triangle with vertex angle 2π/n radians (360/n degrees). The equal angles measure ½(π-2π/n) radians=½π-π/n radians (90-180/n degrees). Since the arcs are congruent, so are the isosceles triangles all round the circle. The angle between two consecutive chords is twice the measure of the equal angles of the isosceles triangles=π-2π/n radians (180-360/n degrees).

The interior angles of an n-sided polygon (regular or irregular) sum to 2n-4 right angles, that is, (2n-4)π/2=nπ-2π radians or 180n-360 degrees. Each interior angle of a regular polygon (all interior angles equal in measure) therefore measures (nπ-2π)/n=π-2π/n radians or (180n-360)/n=180-360/n degrees. But this is the same measure as the angle between two consecutive chords formed by the congruent arcs. Therefore, the chords form the sides of the regular polygon.

by Top Rated User (1.1m points)

Related questions

0 answers
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!

Most popular tags

algebra problems solving equations word problems calculating percentages math problem geometry problems calculus problems math fraction problems trigonometry problems rounding numbers simplifying expressions solve for x order of operations probability algebra pre algebra problems word problem evaluate the expression slope intercept form statistics problems factoring polynomials solving inequalities 6th grade math how to find y intercept equation of a line sequences and series algebra 2 problems logarithmic equations solving systems of equations by substitution dividing fractions greatest common factor square roots geometric shapes graphing linear equations long division solving systems of equations least to greatest dividing decimals substitution method proving trigonometric identities least common multiple factoring polynomials ratio and proportion trig identity precalculus problems standard form of an equation solving equations with fractions http: mathhomeworkanswers.org ask# function of x calculus slope of a line through 2 points algebraic expressions solving equations with variables on both sides college algebra domain of a function solving systems of equations by elimination differential equation algebra word problems distributive property solving quadratic equations perimeter of a rectangle trinomial factoring factors of a number fraction word problems slope of a line limit of a function greater than or less than geometry division fractions how to find x intercept differentiation exponents 8th grade math simplifying fractions geometry 10th grade equivalent fractions inverse function area of a triangle elimination method story problems standard deviation integral ratios simplify systems of equations containing three variables width of a rectangle percentages area of a circle circumference of a circle place value solving triangles parallel lines mathematical proofs solving linear equations 5th grade math mixed numbers to improper fractions scientific notation problems quadratic functions number of sides of a polygon length of a rectangle statistics zeros of a function prime factorization percents algebra 1 evaluating functions derivative of a function equation area of a rectangle lowest common denominator solving systems of equations by graphing integers algebra 2 diameter of a circle dividing polynomials vertex of a parabola calculus problem perpendicular lines combining like terms complex numbers geometry word problems converting fractions to decimals finding the nth term range of a function 4th grade math greatest to least ordered pairs functions radius of a circle least common denominator slope unit conversion solve for y calculators solving radical equations calculate distance between two points area word problems equation of a tangent line multiplying fractions chemistry binomial expansion place values absolute value round to the nearest tenth common denominator sets set builder notation please help me to answer this step by step significant figures simplifying radicals arithmetic sequences median age problem trigonometry graphing derivatives number patterns adding fractions radicals midpoint of a line roots of polynomials product of two consecutive numbers limits decimals compound interest please help pre-algebra problems divisibility rules graphing functions subtracting fractions angles numbers discrete mathematics volume of a cylinder simultaneous equations integration probability of an event comparing decimals factor by grouping vectors percentage expanded forms rational irrational numbers improper fractions to mixed numbers algebra1 matrices logarithms how to complete the square mean statistics problem analytic geometry geometry problem rounding decimals 5th grade math problems solving equations with variables solving quadratic equations by completing the square simplifying trigonometric equation using identities
87,441 questions
99,039 answers
2,422 comments
16,938 users