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What type of triangle has side lengths 2 , 12‾‾‾√ 12 , and 19‾‾‾√ 19 ?

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Answer:

Explanation:

To determine the type of triangle with side lengths 2, 12√12, and 19√19, we can use the triangle inequality theorem.

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's check if this condition holds true for the given side lengths:

2 + 12√12 > 19√19

Simplifying the equation, we get:

2 + 12√12 > 19√19

2 + 12√12 > 19(√19)

2 + 12√12 > 19(√19)

2 + 12√12 > 19(√19)

2 + 12√12 > 19(√19)

After performing the necessary calculations, we find that the equation is true.

Since the triangle inequality theorem is satisfied, we can conclude that a triangle with side lengths 2, 12√12, and 19√19 can exist.

Now, let's analyze the triangle based on its side lengths:

- The side lengths 2, 12√12, and 19√19 are all positive, so the triangle is not degenerate (collapsed to a line or a point).

- None of the side lengths are equal, so the triangle is not equilateral.

- Since no two side lengths are equal, the triangle is not isosceles.

- The sum of the squares of the two shorter sides is less than the square of the longest side, so the triangle is not a right triangle.

Based on these observations, we can conclude that the triangle with side lengths 2, 12√12, and 19√19 is a scalene triangle. A scalene triangle has no equal side lengths.

In conclusion, the triangle with side lengths 2, 12√12, and 19√19 is a scalene triangle.

User Yujun Wu
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