Q. 45.0( 3 Votes )

A circle touches

Answer :


Given that a circle touches all the three sides of a right angled ΔABC.


We have to prove that radius of a circle =


Proof:


Let radius be r and the centre of the circle touching all the sides of a triangle be I i.e. incentre.


In the figure,


ID = IE = IF = r


Since ΔABC is a right angled triangle, B = 90°.


Also ID BC and AB BC.


ID || AB and ID || FB


Similarly, IF || BD


IFBD is a parallelogram.


FB = ID = r and BD = IF = r … (1)


Parallelogram IFBD is a rhombus.


Since B = 90°, parallelogram IFBD is a square.


Now AE = AF


AE = AB – FB


= AB – r [From (1)] … (2)


And CE = CD


CE = BC – BD


= BC – r [From (1)] … (3)


Now, AC = AE + CE,


AC = AB – r + BC – r


AC = AB + BC – 2r


2r = AB + BC – AC


r =


The radius of a circle is .


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