Question

What is the radius of the circle inscribed in triangle $abc$ if $ab = 12, ac=14, bc=16$? express your answer in simplest radical form?

Answer 1

The triangle given above is a scalene triangle as non of its given sides are equal. To determine the area of the triangle, we use the Heron's formula which is expressed as below,
    A = sqrt ((s)(s - a)(s - b)(s - c))

where a, b, and c are the lengths of the sides and s is the semiperimeter ((a+b+c)/2)

The area of the triangle can also be computed through the equation, A = rs
where r is the radius of the inscribed circle. Equating both areas,
     A = sqrt ((s)(s-a)(s-b)(s-c)) = rs
Divide the equation by s,
      r = sqrt ((s)(s-a)(s-b)(s-c)) / s


Solving for s,
    s = (12+14+16)/2 = 21

Substituting to the equation for r,
    r = sqrt ((21)(21-12)(21-14)(21-16)) 21
    r = sqrt 15
    r = 3.87

ANSWER: r = 3.87

Answer 2

Answer:sqrt15

Look at the picture