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

Question

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

James Atkins · Accepted Answer

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.

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)#

Noah Atkinson · Answer

undefined The radius of the triangle's inscribed circle can be calculated using the formula r = A / s, where A is the area of the triangle and s is the semiperimeter of the triangle. The area of the triangle can be calculated using Heron's formula: A = √(s(s - a)(s - b)(s - c)), where a, b, and c are the lengths of the sides of the triangle, and s = (a + b + c) / 2 is the semiperimeter. Plugging in the values a = 5, b = 9, c = 4 into the formula, we get s = 9, A = √(9(9 - 5)(9 - 9)(9 - 4)) = 6√5. Therefore, the radius of the inscribed circle is r = 6√5 / 9 = 2√5 / 3.