Apply the Triangle Inequality Theorem to determine whether sides with lengths of 4 inches, 8 inches, and 9 inches form a triangle.(1 point)
Responses
Yes, all of the conditions are met.
Yes, all of the conditions are met.
No, because two out of three conditions are met.
No, because two out of three conditions are met.
Yes, the majority of the conditions are met.
Yes, the majority of the conditions are met.
No, all of the conditions are not met.
The Triangle Inequality 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 for all three combinations of sides. Let's check whether the given lengths of 4 inches, 8 inches, and 9 inches satisfy this condition:
1. \( 4 + 8 > 9 \) (This condition is met, as \( 12 > 9 \))
2. \( 4 + 9 > 8 \) (This condition is met, as \( 13 > 8 \))
3. \( 8 + 9 > 4 \) (This condition is met, as \( 17 > 4 \))
Since all three conditions are met, the side lengths given can indeed form a triangle. So the correct response is:
Yes, all of the conditions are met.
Apply the Triangle Inequality Theorem to determine whether sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches form a triangle.(1 point)
Responses
Yes, because the sum of any two sides is greater than the third side.
Yes, because the sum of any two sides is greater than the third side.
No, because the sum of any two sides is greater than the third side.
No, because the sum of any two sides is greater than the third side.
No, because the sum of any two sides is less than the third side.
No, because the sum of any two sides is less than the third side.
Yes, because the sum of any two sides is less than the third side
To determine whether sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches form a triangle, we again apply the Triangle Inequality Theorem, which requires that the sum of the lengths of any two sides must be greater than the length of the third side. Let's verify each condition:
1. \( 4.24 + 9.89 > 8.75 \) (This condition is met, as \( 14.13 > 8.75 \))
2. \( 4.24 + 8.75 > 9.89 \) (This condition is met, as \( 12.99 > 9.89 \))
3. \( 9.89 + 8.75 > 4.24 \) (This condition is met, as \( 18.64 > 4.24 \))
All three conditions are satisfied, indicating that the given side lengths can indeed form a triangle. The correct response is:
Yes, because the sum of any two sides is greater than the third side.
Apply the Triangle Inequality Theorem to determine whether a triangle can be formed from sides of lengths 7 millimeters, 8 millimeters, and 9 millimeters.(1 point)
Responses
Yes, because the sum of any two sides is less than the third side.
Yes, because the sum of any two sides is less than the third side.
No, because the sum of any two sides is greater than the third side.
No, because the sum of any two sides is greater than the third side.
No, because the sum of any two sides is less than the third side.
No, because the sum of any two sides is less than the third side.
Yes, because the sum of any two sides is greater than the third side.
To determine whether a triangle can be formed from sides of lengths 7 millimeters, 8 millimeters, and 9 millimeters, we must check each pair of sides to see if their sum is greater than the length of the remaining side, according to the Triangle Inequality Theorem:
1. \( 7 + 8 > 9 \) (This condition is met, as \( 15 > 9 \))
2. \( 7 + 9 > 8 \) (This condition is met, as \( 16 > 8 \))
3. \( 8 + 9 > 7 \) (This condition is met, as \( 17 > 7 \))
All three conditions are satisfied, so the given side lengths can indeed form a triangle. The correct response is:
Yes, because the sum of any two sides is greater than the third side.
Apply the Triangle Inequality Theorem to determine whether sides with lengths of 5 inches, 10 inches, and 15 inches form a triangle.(1 point)
Responses
Yes, because the sum of 5+10 is less than or equal to 15.
Yes, because the sum of 5 plus 10 is less than or equal to 15.
Yes, because the sum of 5+10 is not greater than 15.
Yes, because the sum of 5 plus 10 is not greater than 15.
No, because the sum of 5+10 is not greater than 15.
No, because the sum of 5 plus 10 is not greater than 15.
Yes, the majority of the conditions are met.
To see if sides with lengths of 5 inches, 10 inches, and 15 inches form a triangle, we again use the Triangle Inequality Theorem, which 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 must check each pair:
1. \( 5 + 10 \) should be greater than \( 15 \)
2. \( 5 + 15 \) should be greater than \( 10 \)
3. \( 10 + 15 \) should be greater than \( 5 \)
Let's check these:
1. \( 5 + 10 = 15 \) which is not greater than the third side \( 15 \) (This condition is not met)
2. \( 5 + 15 = 20 \) which is greater than \( 10 \) (This condition is met)
3. \( 10 + 15 = 25 \) which is greater than \( 5 \) (This condition is met)
For a set of lengths to form a triangle, all three conditions must be satisfied. Because the sum of the lengths of 5 inches and 10 inches equals the length of the third side (15 inches), which does not satisfy the strict inequality required by the Triangle Inequality Theorem, these side lengths cannot form a triangle.
The correct response is:
No, because the sum of 5+10 is not greater than 15.
Apply the Triangle Inequality Theorem to determine whether sides with lengths of 3 inches, 4 inches, and 9 inches form a triangle.(1 point)
Responses
No, because 3+4 is less than 9.
No, because 3 plus 4 is less than 9.
Yes, because 3+4 is less than 9.
Yes, because 3 plus 4 is less than 9.
Yes, the majority of the conditions are met.
Yes, the majority of the conditions are met.
Yes, because the sum of any two sides is less than the third side.
The Triangle Inequality 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's apply this theorem to the given lengths of 3 inches, 4 inches, and 9 inches to see if they can form a triangle:
1. \( 3 + 4 > 9 \) (This condition is not met, as \( 7 \) is not greater than \( 9 \))
2. \( 3 + 9 > 4 \) (This condition is met, as \( 12 > 4 \))
3. \( 4 + 9 > 3 \) (This condition is met, as \( 13 > 3 \))
Even though two out of three conditions are met, the Triangle Inequality Theorem requires that all three conditions must be met for three lengths to be able to form a triangle. Since the sum of the lengths of the two shorter sides (3 inches and 4 inches) is not greater than the length of the third side (9 inches), these lengths cannot form a triangle.
The correct response is:
No, because 3+4 is less than 9.