✒️[tex]\large{\mathcal{ANSWER}}[/tex]
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- This alternative is false.
It's because?
- Because for a triangle to exist, it needs to meet 2 conditions, so let's remember the conditions for a triangle to exist.
What conditions are these?
- A triangle will only exist under the following conditions:
1. One of its sides must be greater than the magnitude of the difference of the other two sides.
2. One of its sides must be less than the sum of the two sides.
• Like this?
For example, let's use as an example the classic example of the triangle that appears in textbooks, the triangle with sides 3, 4 and 5. It will exist, because:
|5 - 4| < 3 > 4 + 5
1 < 3 > 9
Notice that 3 is greater than 1 and less than 9. So it meets both conditions. Only one side is not enough to meet these two conditions, the 3 sides must be greater than the difference between the 2 sides and smaller than the sum of the other sides.
•Knowing this, let us prove that it is not enough for a triangle to have all different sides.
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Let's take, for example, a triangle that has sides 4, 14 and 9.
•This triangle will never exist, because:
•Testing the existence condition with side 4:
|14 – 9| < 4 < 14+ 9
|5| < 4 < 23
just with this example, we can prove that this triangle would not exist, given that 4 is less than 5 and less than 23, that is, it meets the second condition (of being less than the sum of the other 2 sides), but it does not meet the first condition (to be greater than the difference of the other 2 sides).
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Therefore, it is clear that this alternative is false, as it is not enough that all sides are different, they must meet the conditions for the existence of a triangle.
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