Two sides of a triangle have lengths 4 and 7. Which of the following can NOT be the length of the third side
Can't see your choices , but .......
Any side of a triangle must be less than the sum of the other two sides
so if your third side is x , where x is a positive number
x < 4+7 AND 7 < 4+x AND 4 < 7+x
x < 11 AND x > 3 AND x > -3 <----- already contained in x > 3
so
3 < x < 11 , that is, your third side must be any number between 3 and 11
So anything outside that domain would not form a triangle
7-4 < x < 7+4
so the 3rd side must be between 3 and 11
To determine which length cannot be the length of the third side of the triangle, we can 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 third side.
In this case, we have two sides with lengths 4 and 7. Let's call the length of the third side "x".
So we need to check the following conditions:
1. Is 4 + 7 > x?
2. Is 4 + x > 7?
3. Is 7 + x > 4?
By simplifying these inequalities, we get:
1. 11 > x
2. 4 + x > 7
3. 7 + x > 4
Now, we can analyze each answer choice to see which lengths satisfy all three conditions:
- Choice A: 3
For 3 to be the length of the third side, it must satisfy all three conditions:
1. 11 > 3 (True)
2. 4 + 3 > 7 (True)
3. 7 + 3 > 4 (True)
So, 3 can be the length of the third side.
- Choice B: 11
For 11 to be the length of the third side, it must satisfy all three conditions:
1. 11 > 11 (False)
The first condition is not satisfied, which means 11 cannot be the length of the third side.
- Choice C: 6
For 6 to be the length of the third side, it must satisfy all three conditions:
1. 11 > 6 (True)
2. 4 + 6 > 7 (True)
3. 7 + 6 > 4 (True)
So, 6 can be the length of the third side.
- Choice D: 2
For 2 to be the length of the third side, it must satisfy all three conditions:
1. 11 > 2 (True)
2. 4 + 2 > 7 (False)
The second condition is not satisfied, which means 2 cannot be the length of the third side.
Therefore, the length that can NOT be the length of the third side is 2.