A triangle has side lengths of 2 units and 8 units. Which of the following could NOT be a possible length of the third side?

Question

7
8
9
10

Answer 1

To determine which lengths could NOT be a possible length of the third side of a triangle with the other two sides measuring 2 units and 8 units, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the lengths of the sides be $a=2$, $b=8$, and $c$ be the length of the third side. The triangle inequalities to consider are:

1. $a+b>c$
2. $a+c>b$
3. $b+c>a$

Substituting $a$ and $b$:

1. $2+8>c$ → $10>c$ → $c<10$
2. $2+c>8$ → $c>6$
3. $8+c>2$ → $c>-6$ (this inequality is always satisfied given positive side lengths)

From these inequalities, we find:

• $c$ must be less than 10.
• $c$ must be greater than 6.

Thus, the range for $c$ is:

$$6<c<10$$

Now let's evaluate the given options:

• 7: This is within the range (6 < 7 < 10).
• 8: This is also within the range (6 < 8 < 10).
• 9: This is within the range (6 < 9 < 10).
• 10: This is NOT within the range (10 is not less than 10).

Therefore, the length that could NOT be a possible length of the third side is 10.