A triangle has sides with lengths of 5, 8, and 3. What is the radius of the triangles inscribed circle?

Question

A triangle has sides with lengths of 5, 8, and 3.  What is the radius of the triangles inscribed circle?

Theodore Abad · Accepted Answer

There exists no triangle with the given measurements because the given measurements form a straight line. If you still want to calculate the are then I have explained below. If #a, b and c# are the three sides of a triangle then the radius of its inscribed circle is given by #R=Delta/s# Where #R# is the radius #Delta# is the are of the triangle and #s# is the semi perimeter of the triangle. The area #Delta# of a triangle is given by #Delta=sqrt(s(s-a)(s-b)(s-c)# And the semi perimeter #s# of a triangle is given by
#s=(a+b+c)/2# Here let #a=5, b=8 and c=3# #implies s=(5+8+3)/2=16/2=8# #implies s=8# #implies s-a=8-5=3, s-b=8-8=0 and s-c=8-3=5# #implies s-a=3, s-b=0 and s-c=5# #implies Delta=sqrt(8*3*0*5)=sqrt0=0# #implies R=0/8=0# Mathematically, since the area comes out to be zero therefore it proves that it is a straight line it is not a triangle, and since there is no triangle so there is no in-circle and thus no in-radius.

Violet Aldrich · Answer

undefined The radius $ r $ of the inscribed circle in a triangle can be found using the formula:

$$ r = \frac{{\text{{Area of the triangle}}}}{{\text{{Semiperimeter of the triangle}}}} $$

where the semiperimeter $ s $ is given by:

$$ s = \frac{{a + b + c}}{2} $$

and $ a $, $ b $, and $ c $ are the lengths of the sides of the triangle.

Using the lengths of the sides of the triangle (5, 8, and 3), we can calculate the semiperimeter $ s $, and then use it along with the area formula to find the radius $ r $.