Question
Gemma wants to draw a triangle with side lengths of 4 inches, 12 inches, and 17 inches. Which statement is true?
This triangle exists because the sum of any two side lengths is greater than the length of the third side.
This triangle exists because the sum of 4 and 12 is less than 17.
This triangle does not exist because the sum of any two side lengths is greater than the length of the third side.
This triangle does not exist because the sum of 4 and 12 is less than 17.
This triangle exists because the sum of any two side lengths is greater than the length of the third side.
This triangle exists because the sum of 4 and 12 is less than 17.
This triangle does not exist because the sum of any two side lengths is greater than the length of the third side.
This triangle does not exist because the sum of 4 and 12 is less than 17.
Answers
GPT-4o mini
To determine whether a triangle with side lengths of 4 inches, 12 inches, and 17 inches can exist, we can apply the triangle inequality theorem. The theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
We can check the three conditions for our side lengths:
1. \(4 + 12 > 17\)
\[16 > 17\] (This is false)
2. \(4 + 17 > 12\)
\[21 > 12\] (This is true)
3. \(12 + 17 > 4\)
\[29 > 4\] (This is true)
Since the first condition \(4 + 12 > 17\) is not satisfied, we can conclude that this triangle does not exist.
Therefore, the correct statement is:
**This triangle does not exist because the sum of 4 and 12 is less than 17.**
We can check the three conditions for our side lengths:
1. \(4 + 12 > 17\)
\[16 > 17\] (This is false)
2. \(4 + 17 > 12\)
\[21 > 12\] (This is true)
3. \(12 + 17 > 4\)
\[29 > 4\] (This is true)
Since the first condition \(4 + 12 > 17\) is not satisfied, we can conclude that this triangle does not exist.
Therefore, the correct statement is:
**This triangle does not exist because the sum of 4 and 12 is less than 17.**