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In a triangle ABC, if AB, BC and AC are the three sides of a triangle, then which of the following statements is necessarily true?
A. AB + BC < AC
B. AB + BC > AC
C. AB + AB = AC
D. \[A{B^2} + A{B^2} = A{C^2}\]

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Last updated date: 20th Jun 2024
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Answer
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Hint: We will go through each of the options one by one and see if it is possible or not in a general triangle ABC and if there is any particular condition to that or not.

Complete answer:
Let us go through each of the four options given to us one by one:-
Option A: AB + BC < AC
Since, we know that in any triangle ABC, the sum of any two sides is always greater than the third side and thus AB + BC < AC is incorrect.
Here in AB + BC < AC, the sum of two sides is shown to be less than the third side.
Hence, option (A) is incorrect always.
Option B: AB + BC > AC
Since, we know that in any triangle ABC, the sum of any two sides is always greater than the third side and thus AB + BC < AC is incorrect.
Here in AB + BC > AC, the sum of two sides is shown to be greater than the third side.
Hence, option (B) is always correct.
Option C: AB + BC = AC
This can only be true if all the points A, B and C lie on the straight line. But since ABC is given to be a triangle, all the points can never be in a straight line. Hence, it is never possible.
Hence, option (C) is always incorrect.
Option D: \[A{B^2} + A{B^2} = A{C^2}\]
We can clearly observe that this is kind of depicting Pythagorean Theorem. We also know that the Pythagorean Theorem is only applicable to right angled triangles.
Here, we are not given anything about any right angle.
Therefore, it is not necessarily true always but only when ABC is a right angled triangle.
Hence, (D) is incorrect.

Hence, the answer is (B).

Note:
The students must note that they may make the mistake of putting D in correct as well when this is a multiple correct option. But they must note that it is only true under the condition of the triangle being right angled. Therefore, we have chosen only the option which is correct no matter what.