A triangle has sides with lengths of 5, 9, and 4. What is the radius of the triangles inscribed circle?

1 Answer
Jan 21, 2016

Notice that #5+4=9#, which means this is not a real triangle, but a segment with length #9# divided in two parts, #5# and #4#.
So, we cannot talk about inscribed circle.

Explanation:

However, it would be educational to know how to solve this problem in general for real triangles.

Assume, we have a triangle with sides #a#, #b# and #c#. If the radius of an inscribed circle is #r#, the area of this triangle is, obviously,
#S = 1/2(a+b+c)*r#

On the other hand, this same area, according to Heron's formula, is equal to
#S = sqrt(p(p-a)(p-b)(p-c))#,
where #p=(a+b+c)/2#.

From this we can derive an equation
#1/2(a+b+c)*r = sqrt(p(p-a)(p-b)(p-c))#

Solving the above for #r#, we obtain the radius of an inscribed circle:
#r = 2sqrt(p(p-a)(p-b)(p-c))/(a+b+c)#