A geometry problem by A Former Brilliant Member

Geometry Level 3

If the side lengths of a triangle are $a,b$ and $c$ , is it possible that $a^2,$ $b^2$ and $c^2$ can be the side lengths of another triangle?

yes no

2 solutions

Marta Reece
Apr 18, 2017

If $a=b=c=2$ , then $a^2=b^2=c^2=4$ . So an equilateral triangle with side 2 satisfies the requirements, which proves that such triangles do exist.

The problem specifies that the triangles must be different, so your solution is not valid. Ed Gray

Edwin Gray - 2 years, 10 months ago

Fixed that.

Marta Reece - 2 years, 10 months ago

Great, what is the fix? Ed

Edwin Gray - 2 years, 10 months ago

The problem specifies that the triangles must be different, so your solution is not relevant. Ed Gray

Edwin Gray - 2 years, 10 months ago

A Former Brilliant Member
Apr 16, 2017

The positive real numbers $a^2, b^2$ and $c^2$ can be the side lengths of a triangle if and only if $a^2+b^2>c^2, b^2+c^2>a^2$ and $c^2+a^2>b^2$ . In other words the sum of the lengths of two sides must be greater than the third side.

Your solution is cogent, but it doesn't really prove that a different triangle exists. It would be preferable to provide an example. Ed Gray

Edwin Gray - 2 years, 10 months ago

A geometry problem by A Former Brilliant Member

2 solutions

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