Triangle Lengths: Can $a,b,c$ Form a Triangle?

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In summary, the Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than the third side. To determine if three given side lengths can form a triangle, you must first check if they satisfy the Triangle Inequality Theorem. If they do, then you can use the Pythagorean Theorem to check if the triangle is a right triangle. Three equal side lengths can form an equilateral triangle, and there is no specific order in which the side lengths should be given. Negative side lengths cannot form a triangle as they must be positive values to satisfy the Triangle Inequality Theorem.
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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.
 
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We have

$a^2+5b^2+4c^2-4ab-4bc=0$

Or $a^2-4ab + 4b^2 + b^2 - 4bc + 4c^2 = 0$

Or $(a-2b)^2 + (b- 2c)^2= 0$

Because a,b,c are real so each part is zero

This gives the solution a = 2b and b = 4c

Or a = 4c, b =2c and so $a > b+ c$

so the answer is no
 

Related to Triangle Lengths: Can $a,b,c$ Form a Triangle?

1. What are the criteria for a, b, c to form a triangle?

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.

2. Can three equal lengths form a triangle?

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.

3. What is the maximum number of triangles that can be formed with given side lengths?

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.

4. Can a triangle have two sides with the same length?

Yes, a triangle can have two sides with the same length. This type of triangle is called an isosceles triangle.

5. How do you determine the type of triangle based on the given side lengths?

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.

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