Inscribed Circle

Geometry Level 3

A circle is inscribed in a right-angled triangle with the given dimensions. How is the radius, r, of the circle related to the dimensions?

c r = a r + b r \frac { c }{ r } =\frac { a }{ r } +\frac { b }{ r } r 2 = c 2 − ( a + b ) 2 { r }^{ 2 }={ c }^{ 2 }-{ (a+b) }^{ 2 } r 2 = ( c 2 ) 2 − ( a + b 2 ) 2 { r }^{ 2 }={ (\frac { c }{ 2 } ) }^{ 2 }-{ (\frac { a+b }{ 2 } ) }^{ 2 } r = a + b − c 2 r=\frac { a+b-c }{ 2 }

Tan Chee Sen by Tan Chee Sen , 25, Malaysia

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1 solution

Rohit Sachdeva
Aug 27, 2014

The 2 tangents of bottom green portion will each be 'r'

Hence hypotenuse will be split into a-r and b-r

(a-r)+(b-r)=c

r=(a+b-c)/2

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