Two sides of a triangle have lengths 4 and 7. Which of the following can NOT be the length of the third side?
A. 4
B. 7
C. 11
D. 5
D 5
the 3rd side must be greater than the difference, and less than the sum, of the 1st two sides
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Well, I would say that the length of the third side cannot be "B. 7" because that would just make it a line segment lying entirely on one side of the triangle, rather than actually forming a triangle. So, the answer is B. 7.
To answer this question, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
In this case, the lengths of two sides of the triangle are given as 4 and 7. Let's check each option to see if it satisfies the triangle inequality:
A. Is 4 a possible length for the third side? To check, we add 4 and 7: 4 + 7 = 11. Since 11 is greater than 4, it is possible, so option A is a valid length.
B. Is 7 a possible length for the third side? We add 4 and 7: 4 + 7 = 11. Since 11 is greater than 7, it is possible, so option B is a valid length.
C. Is 11 a possible length for the third side? We add 4 and 7: 4 + 7 = 11. Since 11 is equal to 11, it is barely possible, so option C is a valid length.
D. Is 5 a possible length for the third side? We add 4 and 7: 4 + 7 = 11. Since 11 is greater than 5, it is possible. Therefore, option D is a valid length as well.
Based on the triangle inequality theorem, all the given options satisfy the conditions and cannot be ruled out as the length of the third side. Therefore, the answer is none of the above (E).