# If a, b and c are the sides of a right angle triangle where C is the hypotenuse. Prove that radius, r of the circle touches the sides of the triangle given by  r=a+b-c                                                                                                         2 Let the circle touches the sides BC, CA, AB of the right triangle ABC at D, E and F respectively, where BC = a, CA = b and AB = c  Then AE = AF and BD = BF. Also CE = CD = r.
i.e., b – r = AF, a – r = BF
or AB = c = AF + BF = b – r + a – r

This gives r=(a+b-c)/2

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Alternative solution:

c is the hypotenuese and a , b are the adjacent sides of the right angle. Therefore a2 + b2 = c2.

Now area of the riangle ABC = Sum of areas of triangles OAB + OBC + OCA

( 1/2 ) ab = (1/2 ) r ( a + b+ +c )     or r = ab / ( a + b + c )   now muliplying  and dividing  RHS  by ( a + b - c ), we get

r = ab ( a + b - c ) / [ (a + b )2 - c2 ]  = a b ( a + b - c ) / [ a2 + b2 + 2ab - a2- b2 ]          since c2 = a2 + b2  .

r = ab ( a + b - c ) / 2ab  = ( a + b - c ) / 2.

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Can u tell me why u've written (1/2)ab = (1/2)r(a+b+c) ?

• -2

area of the riangle ABC =  ( 1/2 ) ab

Also, Sum of areas of triangles OAB + OBC + OCA = [(1/2) X AB X OF] + [(1/2) X BC X OD] + [(1/2) X AC X OE]

= [(1/2) X c X r] + [(1/2) X a X r] + [(1/2) X b X r]  (here, AB=c, BC=a, AC=b and OF= OD= OE = r)

= (1/2) (r) (a+ b+ c)  (Taking out common as 1/2 X r)

But we know that the area of the riangle ABC = Sum of areas of triangles OAB + OBC + OCA

So, ( 1/2 ) ab = (1/2 ) r ( a + b+ +c )

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• 2
Can anyone tell me dat why CE=CD=r?
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Let the circle touch the sides..
BC,CA,AB of the right triangle ABC at D,E and F and now
BC=a
CA=b and
AB=c
then.. AF=AE And BF = BF
now, b - r = AF, a - r = BF
that is r= ( a+b-c)/2
• -2
Why CE=CD=r ?
• -4
sorry i don't know.
• -4
Ce=cd becoz that is a square
• -1
Here, O is the incentre of the triangle. Thus, 'r' is the inradius of the circle. Multiply the numerator and the denominator by (a+b - c), we have, • -2
sarayu.. on which diagram u r explaining
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why CE=CD=r

• 0
Why CD=r?
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Because odce is a square as shown in fig Equal adjacent sides with 90 degree angle • 0
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