Apply the Triangle Inequality Theorem and determine which option could be the third side of a triangle if the first two sides measure 4 and 6.(1 point)
Responses
1
1
9
9
2
2
11
pick one
Given two sides x and y, with x <= y, the third side z must satisfy
y-x < z < y+x
so, if two sides are 4 and 6, the third side z must satisfy
6-4 < z < 6+4
2 < z < 10
Based on the Triangle Inequality Theorem, if the first two sides measure 4 and 6, the third side must satisfy 2 < z < 10. Therefore, any option that falls within this range could be the third side of a triangle.
Option 1: 1
This option does not satisfy 2 < z < 10.
Option 2: 9
This option satisfies 2 < z < 10.
Option 3: 2
This option satisfies 2 < z < 10.
Option 4: 11
This option does not satisfy 2 < z < 10.
Based on this analysis, both options 2 and 3, which are 9 and 2, respectively, could be the third side of a triangle if the first two sides measure 4 and 6.
To apply the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Option 1: 1
4 + 1 = 5 (not greater than 6)
6 + 1 = 7 (greater than 4)
1 + 6 = 7 (greater than 4)
Option 1 does not satisfy the Triangle Inequality Theorem.
Option 2: 9
4 + 9 = 13 (greater than 6)
6 + 9 = 15 (greater than 4)
9 + 6 = 15 (greater than 4)
Option 2 satisfies the Triangle Inequality Theorem.
Option 3: 2
4 + 2 = 6 (equal to 6)
6 + 2 = 8 (greater than 4)
2 + 6 = 8 (greater than 4)
Option 3 satisfies the Triangle Inequality Theorem.
Option 4: 11
4 + 11 = 15 (greater than 6)
6 + 11 = 17 (greater than 4)
11 + 6 = 17 (greater than 4)
Option 4 satisfies the Triangle Inequality Theorem.
Therefore, options 2 and 4 could be the third side of a triangle if the first two sides measure 4 and 6.