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 Incentroid (Posted on 2019-02-24)
The segment formed by joining the incenter and centroid of a certain right triangle is parallel to one of its legs and has a length of 1 unit.

Find the perimeter of this triangle.

 No Solution Yet Submitted by Danish Ahmed Khan Rating: 3.0000 (1 votes)

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 Solution Comment 2 of 2 |

Let AHB be the right triangle with a, h, and b
the lengths of sides HB, BA, and AH
respectively ( h^2 = a^2 + b^2 ). Let I denote
the incenter and M the centroid. We will
assume that b > a. Let r denote the inradius
with r = ( a + b - h ) / 2.

IM parallel to AH ==> a/3 = r
==> a = 3b/4 & h = 5b/4

|IM| = b/3 - r     ==> |IM| = b/12

Perimeter = ( a + b + h ) / |IM|
= ( 3b/4 + b + 5b/4 ) / b/12
= 36

Edited on March 4, 2019, 11:51 pm
 Posted by Bractals on 2019-03-04 23:47:55

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