Try being a Hero (part 2)
Geometry
Level
pending
Consider the smallest possible non-degenerate scalene triangle whose sides are integer lengths in meters.
What is this triangle's area in square meters, to 3 decimal places?
The answer is 2.905.
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1 solution
The difference between the two larger sides of a scalene triangle with integer side lengths has to be a minimum of 1. So for the triangle to be non-degenerate, the smallest side has to have a length of at least 2.
Fix the smallest side at 2. The smallest possible set of values for the two larger sides are therefore 3 and 4. So our triangle has side lengths of 2, 3 and 4 meters. The semi-perimeter is ( 2 + 3 + 4 ) / 2 = 4 . 5 meters.
The area in square meters, by Hero's formula, is therefore ( 4 . 5 ) ( 4 . 5 − 4 ) ( 4 . 5 − 3 ) ( 4 . 5 − 2 ) square meters, which equals 2 . 9 0 5 square meters.