Question

in a triangle abc angle abc equal to 90 degree a circle inscribed in a triangle with radius r.a,b,c are the length of BC AC and ab respectively then prove that 2r=a+b-c

Answer 1

Given: Right angled triangle ACB

           Radius of inscribed circle = r

           BC= a, AC = b, AB = c

To prove: 2r = a + b - c

Proof:  

Since the points produce tangents to the circle,

AP = AQ

CP = CR

BR = BQ

Given,

AB = c

AC = b

BC = a

Since CPOR is a square hence,

CP = CR = r

BR = BC - CR = a - r

BQ = BR = a - r

similarly,

AP = AC - CP = b - r

AQ = AP =  b - r

Now,

AB = c

AQ + BQ = c

b - r + a - r = c

b + a - 2r = c

2r = a + b – c

Hence proved.

Answer 2

2r = c + a - b if in a triangle abc angle abc equal to 90 degree a circle inscribed in a triangle with radius r.a,b,c are the length of BC AC and ab respectively

Step-by-step explanation:

Let draw perpendicular from center O at AB  & BC and mark them P & Q

OP = OQ = Radius = r

as all angles of opbQ are 90 degrees

hence BP = BQ = r

Let say AC touches circle at M

AC = AM + MC

AM = AP  & CM = CQ  ( equal tangent)

AC = AP  + CQ

=> AC = AB - BP  + BC  - BQ

=> AC = AB - r + BC - r

=> 2r  = AB + BC - AC

=> 2r = c + a - b

to get 2r = a + b - c   triangle should be right angles at C

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