Q. 45.0( 4 Votes )

Prove the radius

Answer :


Let ABC be an equilateral triangle, its incircle and circumcircle are drawn with center O, let the radius of incircle be 'r' and circumcircle be 'R'


To prove: R = 2r


Construction: Draw OA BC and OB AC


Proof:


In ΔOAC and ΔOBC


OAC = OBC [Both 90°]


OC = OC [Common]


OA = OB [Radii of incircle]


ΔOAC ΔOBC [By Right Angle - Hypotenuse - Side Criterion]


⇒ ∠OCA = OCB [Corresponding parts of congruent triangles are equal]


But,


C = 60° [Angle of equilateral triangle]


⇒ ∠OCA + OCB = 60°


⇒ ∠OCA + OCA = 60°


2OCA = 60°


⇒ ∠OCA = 30°


Now, In right-angled triangle OAC





[As, OA = radius of incircle = r and OC = radius of circumcircle = R]



R = 2r


Hence Proved.


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