Lower bound = minimum
Algebra
Level
3
If are sides of a triangle, what is the minimum value of
The answer is 1.5.
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1 solution
To have the least ratios between any two sides, they must be equilateral, in the ratio 1:1:1. Consider a non-equilateral triangle with sides 3, 4, and 5. We may have the ratios 3:4, 4:5, and 3:5 which are less than 1:1, but then consider the ratios 5:3, 4:3 and 5:4. When added together the six ratios of the equilateral triangle add up to 6 and the ratios of the non-equilateral triangle add up to 6.4. Any other triangle whose angles are not all equal will have a ratio sum greater than 6 . If we plug in 1 for all variables, you would get 3 x 1/2 = 3/2. It would still be 3/2 for any other side length because it would simplify to 3/2.