A right angled triangle is such that its hypotenuse is equal to 65 centimeters (cm) and its inradius is equal to 12 cm. The length of the other two sides, when expressed in cm, are positive integers.

Determine the perimeter of the triangle.

NOTE: The inradius is defined as the radius of the inscribed circle.

- Write the inradius as expression of the sides a, b and c:
  r=ab/(a+b+c)
  To get the formula express the triangle's area as sum of the three triangles areas that are formed when connecting the incircle midpoint with the triangles corners

- Write a, b and c in parametric form so that integers come out using the school formula for pythagorean triples:
a=2st, b=s^2-t^s, c=s^2+t^2

- Use the above to calculate r:
r=t(s-t)

- Express 65 as sum of squares s^2 and t^2. (s,t)=(8,1) and (s,t)=(7,4) work. Insert into the radius formula an you'll get r=7 and r=12 respectively, so it must be (s,t)=(7,4). Follows a=56, b=33 and hence a+b+c=154.

Maybe there's an easier way, don't know.

I wonder if there is a parametric formula for the length of arbitrary integer-sided triangles with integer-length inradius, outer radius etc. etc.

Posted by JLo on 2006-10-31 07:42:28