What is the radius of the circle inscribed in triangle $abc$ if $ab = ac=7, bc=6$? express your answer in simplest radical form.

We are given

length of sides are

AB=7

AC=7

BC=6

now, we can use formula of area of triangle

where

A is area of triangle

r is radius of in-circle

s is semi-perimeter of triangle

step-1: Finding area of triangle

We can use

Heron's formula

a , b and c are values of sides

now, we can plug values

step-2: Finding radius(r)

we got

now, we can find radius

............Answer

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

e.

step-by-step explanation:

because there are always 10 numbers in a phone number.