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Bot GPT 3.5 The Triangle Inequality Theorem states that for a triangle with side lengths a, b, and c, the sum of any two sides must be greater than the third side.
Using this theorem, we can determine which lengths cannot be the measure of the third side for a triangle with side lengths of 4 cm and 9 cm.
For the first question, the possible measures of the third side are:
6 cm - (4 + 6) > 9
5 cm - (4 + 5) < 9
8 cm - (4 + 8) > 9
10 cm - (4 + 10) > 9
The second choice, 5 cm, is the only length that does not satisfy the Triangle Inequality Theorem and cannot be the measure of the third side.
For the second question, the possible measures of the third side are:
11 - (4 + 6) > 9
9 - (4 + 9) < 6
1 - (4 + 1) < 6
2 - (4 + 2) < 6
The third choice, 1, is the only length that does not satisfy the Triangle Inequality Theorem and cannot be the measure of the third side.
For the third question, we need to determine if the group of side measures will form a triangle:
8, 1, 7 - (1 + 7) > 8
12, 10, 22 - (10 + 12) < 22
9, 4, 3 - (3 + 4) > 9
2, 7, 6 - (2 + 6) > 7
The first and fourth choices, 8, 1, 7 and 2, 7, 6, both satisfy the Triangle Inequality Theorem and will form a triangle.
Therefore, the correct answers are:
- In the first question, 5 cm cannot be the measure of the third side.
- In the second question, 1 cm cannot be the measure of the third side.
- In the third question, the group of side measures 8, 1, 7 will form a triangle.
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