Question

proof a circle is divided into n congruent arcs (n ?? 3), the tangents drawn at the endpoints of these arcs form a regular polygon.

Answer 1

Prove: A circle is divided into 'n' congruent arcs (assume n=3), the tangents drawn at the endpoints of these arcs form a regular Polygon.

Assuming n=3, Let us First Draw the Figure. When the three tangents drawn at the endpoints of the 3 equiangular arcs intersect each other THEN we may get a triangle (Refer diagram below)

                                     

Given: 3 tangents intersecting each other at Points A, B and C intersect the circle at Points D, E and F respectively, Here

To Prove: is a regular polygon i.e. an equilateral triangle

Proof:

.........................(given)......(1)

Now   ....(total sum of all consecutive arcs of a circle is 360 )

............(From 1) .............(2)

In , AC, BC and AB are tangents intersecting the circle at points D, F and E

Now, the center of the circle is A₁ . The Property of the tangent states that A tangent is perpendicular to the radius.

(Refer Figure).............(3)

In , ....... (From 3)

and ...........(The measure of the central angle is equal to the measure of the arc that it intercepts )

(Since, The sum of all angles of a quadrilateral is 360 degree ) ........................(4)

Similarly we can prove in the same manner like above, .....(5)

and

......(6)

From 4, 5 and 6

is an equiangular. Therefore it is an equilateral triangle ( Each angle ) which is a REGULAR POLYGON.....(Since a regular polygon is a closed figure whose sides and angles are both equal )

Hence, A circle which is divided into 'n' congruent arcs (assume n=3), Then the tangents drawn at the endpoints of these arcs form a regular Polygon. (Thus proved)