# Determine the circle radius if the circle is inscribed in a triangle wich has the sides lenghts of 13, 14, and 15 .

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Let A be the area of the triangle

P be the perimeter of the tirangle

and r is the radius of the circlr:

Then, we know that:

A = r * P/2

Let us caclculateL

P = 13 + 14 + 15 = 42

==> p/2 = 42/2 = 21

Now for A:

A =sqrt[ p/2 (p/2- 13)(p/2 - 14)(p/2 - 15)]

= sqrt[21*8*7*6]

= 84

Then:

A = r* p/2

84 = r * 21

==> r= 84/21= 4

**Then the radius = 4 units**

We have area A of the triangle = sqrt{s(s-a)(s-b)s-c)}

a = 1, b =14, c 15

Therefore s= (13+14+15)/2 = 42/2 = 21.

Area = sqrt{21(21-13)(21-14)21-15)} = sqr(21*8*7*6} = 84.

Also area of the triangle = (1/2) (a+b+c)r = 84.

Therefore , (1/2)(42)r = 84

r = 84*2/42 = 4.

If we know the lengths of the triangle sides, we could calculate the radius with the help of the following formula: R=S/p, where R-radius, S-aria of the triangle and p-semi-perimeter of the triangle.

p=(13+14+15)/2=21

S= sq root [21(21-13)(21-14)(21-15)]=84

R=84/21=4

We have to find the radius of the circle inscribed in a triangle with sides of length 13, 14 and 15.

Now we have the area of a triangle given as sqrt [ s ( s-a) (s-b) (s-c)] and it is also equal to r*s where s is the semi perimeter and r is the radius of the inscribed circle.

So r*s = sqrt [ s ( s-a) (s-b) (s-c)]

r = sqrt [ s ( s-a) (s-b) (s-c)] / s = sqrt [( s-a) (s-b) (s-c)]/ s ].

As the sides are 13, 14 and 15, the semi perimeter s = 21 .

So substituting we get r = sqrt [ 8* 7* 6 / 21] = sqrt 16 = 4.

**So the required radius is 4**