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"Seeing" is the Key, Especially in the Sun

After several number theoretic problems, it is necessary to pose a geometric problem. So here goes....

The perimeter of a right triangle is 324 cm and its hypotenuse is 135 cm. Find the radius of the circle that can be inscribed in the triangle.

Source: M. Teitelbaum, "Simpler solution," Mathematics Teacher, October 1987

Hint: The solution path can be either complicated or straight-forward....which approach are you trying?

For the straight-forward approach, draw the picture (i.e. circle incribed in a right triangle), then stand back and look at it....what geometrical "laws" do you see in action?

The graphic (and title) is a hint as well.

Solution Commentary: For the more complicated approach, consult Problem #27 in the Mathematics Teacher" (December 1986).

Teitelbaum suggests one focus on the circle breaking the triangle into three examples of pairs of tangents to a circle from an external point. Remember, these pairs of tangents are congruent. Is this enough to get you on a good track....you should get an answer of a radius = 27 cm.

Solution proposed by D.K. (Fife, WA): Suppose right triangle ABC has right angle B. Use the perimeter of 324 cm, Pythagoreans theorem and trial and error to find AB = a = 81, BC = b = 108 and we know the AC = c = 135. Suppose x, y, and z are points of tangency to circle with center i on AC, AB, and BC respectively. As we know points of tangency are perpendicular to the radius, they can be used as heights of triangles.

Since all radii are congruent, xi = yi = zi = r (radius). Divide triangle ABC into three smaller triangles: AiB, BiC and AiC. Then area of triangle ABC = sum of areas of triangles AiB, BiC, and AiC. Or,

0.5 ab = 0.5 a (yi) + 0.5 b (zi) + 0.5 c (xi)
0.5 ab = 0.5 ar + 0.5 br + 0.5 cr
0.5 ab = 0.5r(a + b + c)
ab = r(a +b +c)
[ab]/[a+b+c] = r
[(81)(108)]/[81+108+135] = r
8748/324 = 27 = r

Note: There is still another "quick" way to solve this problem using properties of tangents to circles.