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anemone
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If $a,\,b$ and $c$ are the sides of a triangle $ABC$, prove that if $a^2+b^2>5c^2$, then $c$ is the length of the shortest side.
MarkFL said:Suppose we assume:
\(\displaystyle a>c\implies a^2>c^2\)
\(\displaystyle b>c\implies b^2>c^2\)
These two conditions also imply:
\(\displaystyle ab>c^2\)
Adding the three implications, we obtain:
\(\displaystyle a^2+ab+b^2>3c^2\)
The triangle inequality implies:
\(\displaystyle a^2+2ab+b^2>c^2\)
And, we are given:
\(\displaystyle a^2+b^2>5c^2\)
Adding these last two, there results:
\(\displaystyle a^2+ab+b^2>3c^2\)
As we initially found, this is the result of assuming both \(a\) and \(b\) are greater than \(c\). :)
kaliprasad said:Hello Mark
You have assumed c being the shortest side and you have taken the given condition as well. So I am not convinced that
the proof is right. You have taken both the condition and assumption and proved it.
If I have missed something kindly let me know.
MarkFL said:I have assumed that \(c\) is the shortest side and shown how it leads to an implication provided both by the given and the triangle inequality. It seems to me this is sufficient. Is it not?
kaliprasad said:But does not prove that this is not true if c is not the shortest side which we need to prove
The length of the shortest side in a triangle can vary depending on the specific triangle. However, in a right triangle, the shortest side is always the side opposite the smallest angle.
To find the length of the shortest side in a triangle, you can use the Pythagorean Theorem. This theorem states that the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. Therefore, you can find the length of the shortest side by subtracting the square of the other two sides from the square of the hypotenuse and then taking the square root of the result.
No, the shortest side in a triangle cannot be longer than the other two sides. This is because in a triangle, the sum of any two sides must be greater than the third side. Therefore, the shortest side must be shorter than the other two sides.
No, the length of the shortest side in a triangle does not always have to be an integer. It can be a decimal or a fraction depending on the specific triangle. However, in a right triangle with integer side lengths, the length of the shortest side will also be an integer.
The length of the shortest side in a triangle does not determine the shape of the triangle. The shape of a triangle is determined by the lengths of all three sides and the angles between them. However, in a right triangle, the length of the shortest side is directly related to the size of the smallest angle, which can affect the overall shape of the triangle.