Question 1170611: The length of the radius of the inscribed circle of the right triangle is 4. If all lengths are natural numbers, find all possible options for the leg lengths.
Answer by greenestamps(13200) (Show Source):
You can put this solution on YOUR website!
Since the side lengths are natural numbers, we are looking at triangles whose side lengths are a Pythagorean Triple.
If the lengths of the legs are a and b with hypotenuse c, then the area is given by

and by

where r is the radius of the inscribed circle.
Since we want the radius of the inscribed circle to be 4, we are looking for Pythagorean Triples in which

or

We can simply look at each Pythagorean Triple and see if the radius of the inscribed circle is 4:
a,b,c ab a+b+c r = ab/(a+b+c)
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3,4,5 12 12 1
5,12,13 60 30 2
7,24,25 168 56 3
8,15,17 120 40 3
9,40,41 360 90 4
11,60,61 660 132 5
All larger Pythagorean Triples will make triangles in which the radius of the inscribed circle is greater than 4, so we don't need to look any further.
We see from the list that there is one primitive Pythagorean Triple (9-40-41) that gives 4 for the radius of the inscribed circle.
We can also see that we can get a radius of 4 by scaling up the 3-4-5 triangle by a factor of 4, or by scaling up the 5-12-13 triangle by a factor of 2.
So there are exactly three right triangles with side lengths that are natural numbers in which the radius of the inscribed circle is 4:
12,16,20
10,24,26
9,40,41
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