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

Share with your friends

Parishkrit Jain answered this

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

View Full Answer

Parishkrit Jain answered this

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.

Antariksh Choudhary answered this

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

-2

Parishkrit Jain answered this

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 )

Nandini answered this

Please give the answer......

Miyala answered this

Can anyone tell me dat why CE=CD=r?

Madhumitha answered this

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

Shreyans answered this

Why CE=CD=r ?

-4

Swasti answered this

sorry i don't know.

-4

Sandhya answered this

Ce=cd becoz that is a square

-1

Sarayu answered this

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

Srinivas Karimnagar answered this

sarayu.. on which diagram u r explaining

Anurag answered this

why CE=CD=r

Prem Kalra answered this

Why CD=r?

Versha Rani answered this

Because odce is a square as shown in fig Equal adjacent sides with 90 degree angle

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

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