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anemone
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Three positive real numbers $a,\,b$ and $c$ are such that $a^2+5b^2+4c^2-4ab-4bc=0$. Can $a,\,b$ and $c$ be the lengths of the sides of a triangle? Justify your answer.
The criteria for a, b, c to form a triangle is that the sum of any two sides must be greater than the third side. In other words, a + b > c, a + c > b, and b + c > a.
Yes, three equal lengths can form a triangle as long as they meet the criteria mentioned above. This type of triangle is called an equilateral triangle.
The maximum number of triangles that can be formed with given side lengths is one. If the given side lengths do not meet the criteria for a triangle, then no triangle can be formed.
Yes, a triangle can have two sides with the same length. This type of triangle is called an isosceles triangle.
The type of triangle can be determined based on the given side lengths by comparing the lengths. If all three sides are equal, it is an equilateral triangle. If two sides are equal, it is an isosceles triangle. If all three sides are different lengths, it is a scalene triangle.