Given that AB¯¯¯¯¯¯¯¯=9 units in length and BC¯¯¯¯¯¯¯¯=12 units in length, how many units is CE¯¯¯¯¯¯¯¯ ?
21
24
15
6
To find the length of CE¯¯¯¯¯¯¯¯, we need to use the triangle inequality theorem. According to this theorem, 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, AB¯¯¯¯¯¯¯¯+BC¯¯¯¯¯¯¯¯>AC¯¯¯¯¯¯¯¯.
Substituting the given values, we have:
9 + 12 > AC¯¯¯¯¯¯¯¯
21 > AC¯¯¯¯¯¯¯¯
Since AC¯¯¯¯¯¯¯¯ represents the length of the entire triangle, CE¯¯¯¯¯¯¯¯ must be shorter than this. Therefore, the length of CE¯¯¯¯¯¯¯¯ cannot be 21, eliminating option 1.
We are left with options 2, 3, and 4.
Let's consider the extreme case where CE¯¯¯¯¯¯¯¯ is as long as possible. This would occur when A, B, and C are collinear, creating a straight line. In this case, CE¯¯¯¯¯¯¯¯ would be equal to the sum of the lengths of AB¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯: 9 + 12 = 21.
Therefore, the length of CE¯¯¯¯¯¯¯¯ cannot be 21, eliminating option 2.
We are now left with options 3 and 4: 15 and 6.
Since AB¯¯¯¯¯¯¯¯ is 9 units long, and BC¯¯¯¯¯¯¯¯ is 12 units long, the length of CE¯¯¯¯¯¯¯¯ must be less than the sum of these two lengths: 9 + 12 = 21.
Therefore, the length of CE¯¯¯¯¯¯¯¯ cannot be 24, eliminating option 3.
The only remaining option is 6. Since 6 is less than the sum of 9 and 12, it satisfies the triangle inequality theorem and can be the length of CE¯¯¯¯¯¯¯¯.
Therefore, the answer is 6.