The circle inside a 3-4-5 triangle

Question

A right-angled triangle has legs of length 3 and 4 and hypotenuse 5.

What is the radius of the circle inscribed in the triangle, the circle that touches all three sides? Find a way to see the answer without heavy algebra.

Accepted Answer

The radius is 1. Method 1: areas. Let the radius be r and the centre be I. Joining I to the three corners divides the triangle into three triangles whose bases are the sides 3, 4 and 5 and whose heights are all r (the radius to a point of tangency is perpendicular to the side). Their areas add up to the whole area: (1/2)(3)r + (1/2)(4)r + (1/2)(5)r = 6r. The whole triangle has area (1/2)(3)(4) = 6. So 6r = 6 and r = 1. Method 2: the right-angle corner. At the right angle, the circle touches both legs, so the centre sits at (r, r) if the right angle is at the origin. The two tangent lengths fr…

The circle inside a 3-4-5 triangle

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