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.

Question

2
11
1
9

Answer 1

To determine if a certain length could be the third side of a triangle, we can use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

So, for a triangle with side lengths of 4 and 6:

4 + 6 = 10

Let's check each option:

1. 2
4 + 2 = 6 (This does not satisfy the theorem as 6 is not greater than 2)
2. 11
4 + 11 = 15 (This satisfies the theorem as 15 is greater than 11)
3. 1
4 + 1 = 5 (This satisfies the theorem as 5 is greater than 1)
4. 9
4 + 9 = 13 (This satisfies the theorem as 13 is greater than 9)

So, the third side of the triangle could be 1 or 9.

Answer 2

choose one answer

Answer 3

The third side of the triangle could be 1.

Answer 4

Apply the Triangle Inequality Theorem and determine which group of side measures will form a triangle.

12, 10, 22
8, 1, 7
2, 7, 6
9, 4, 3

Answer 5

To determine which group of side measures will form a triangle, we need to apply the Triangle Inequality Theorem. The 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 check each group of side measures:

1. 12, 10, 22
12 + 10 = 22 (This satisfies the theorem as 22 is greater than 22, but it fails to satisfy the other two combinations, making it unable to form a triangle)

2. 8, 1, 7
8 + 1 = 9 (This satisfies the theorem as 9 is greater than 7)
1 + 7 = 8 (This satisfies the theorem as 8 is greater than 7)
8 + 7 = 15 (This satisfies the theorem as 15 is greater than 8)
Therefore, the lengths 8, 1, 7 can form a triangle.

3. 2, 7, 6
2 + 7 = 9 (This satisfies the theorem as 9 is greater than 6)
2 + 6 = 8 (This satisfies the theorem as 8 is greater than 7)
7 + 6 = 13 (This satisfies the theorem as 13 is greater than 2)
Therefore, the lengths 2, 7, 6 can form a triangle.

4. 9, 4, 3
9 + 4 = 13 (This satisfies the theorem as 13 is greater than 3)
4 + 3 = 7 (This satisfies the theorem as 7 is greater than 4)
9 + 3 = 12 (This satisfies the theorem as 12 is greater than 4)
Therefore, the lengths 9, 4, 3 can form a triangle.

Therefore, the combinations that can form a triangle are:
- 8, 1, 7
- 2, 7, 6
- 9, 4, 3

Answer 6

choose one

Answer 7

The group of side measures that will form a triangle is 8, 1, 7.

Answer 8

Two sides of a triangle measure 3.7 and 8.2. Apply the Triangle Inequality Theorem to find a possible measure of the third side.

4.2
5.5
3.5
4.5

Answer 9

To find a possible measure of the third side of the triangle, we need to apply the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given that the two sides measure 3.7 and 8.2:

3.7 + 8.2 = 11.9

Now, let's check each option:

1. 4.2
3.7 + 4.2 = 7.9 (This satisfies the theorem as 7.9 is less than 8.2)
4.2 + 8.2 = 12.4 (This satisfies the theorem as 12.4 is greater than 3.7)

2. 5.5
3.7 + 5.5 = 9.2 (This satisfies the theorem as 9.2 is less than 8.2)
5.5 + 8.2 = 13.7 (This satisfies the theorem as 13.7 is greater than 3.7)

3. 3.5
3.7 + 3.5 = 7.2 (This satisfies the theorem as 7.2 is less than 8.2)
3.5 + 8.2 = 11.7 (This satisfies the theorem as 11.7 is greater than 3.7)

4. 4.5
3.7 + 4.5 = 8.2 (This satisfies the theorem as 8.2 is equal to 8.2)

Therefore, the possible measure of the third side of the triangle is 4.5.