Question
Apply the Triangle Inequality Theorem to determine which three side lengths form a triangle.(1 point)
Responses
10, 20, 15
8, 8, 20
8, 4, 12
20, 10, 30
Responses
10, 20, 15
8, 8, 20
8, 4, 12
20, 10, 30
Answers
GPT 3.5
To determine whether a triangle can be formed, 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.
Let's apply this theorem to each set of side lengths:
1) 10, 20, 15:
10 + 15 = 25 (greater than 20)
10 + 20 = 30 (greater than 15)
20 + 15 = 35 (greater than 10)
All the pairs of side lengths satisfy the inequality, so a triangle can be formed.
2) 8, 8, 20:
8 + 8 = 16 (smaller than 20)
8 + 20 = 28 (greater than 8)
8 + 20 = 28 (greater than 8)
Only the pair 8 + 20 satisfies the inequality, so a triangle can be formed.
3) 8, 4, 12:
8 + 4 = 12 (equal to 12)
8 + 12 = 20 (greater than 4)
4 + 12 = 16 (greater than 8)
Two out of three pairs satisfy the inequality, so a triangle can be formed.
4) 20, 10, 30:
20 + 10 = 30 (equal to 30)
20 + 30 = 50 (greater than 10)
10 + 30 = 40 (greater than 20)
Two out of three pairs satisfy the inequality, so a triangle can be formed.
In conclusion, three side lengths that form a triangle are:
10, 20, 15
8, 4, 12
20, 10, 30
Let's apply this theorem to each set of side lengths:
1) 10, 20, 15:
10 + 15 = 25 (greater than 20)
10 + 20 = 30 (greater than 15)
20 + 15 = 35 (greater than 10)
All the pairs of side lengths satisfy the inequality, so a triangle can be formed.
2) 8, 8, 20:
8 + 8 = 16 (smaller than 20)
8 + 20 = 28 (greater than 8)
8 + 20 = 28 (greater than 8)
Only the pair 8 + 20 satisfies the inequality, so a triangle can be formed.
3) 8, 4, 12:
8 + 4 = 12 (equal to 12)
8 + 12 = 20 (greater than 4)
4 + 12 = 16 (greater than 8)
Two out of three pairs satisfy the inequality, so a triangle can be formed.
4) 20, 10, 30:
20 + 10 = 30 (equal to 30)
20 + 30 = 50 (greater than 10)
10 + 30 = 40 (greater than 20)
Two out of three pairs satisfy the inequality, so a triangle can be formed.
In conclusion, three side lengths that form a triangle are:
10, 20, 15
8, 4, 12
20, 10, 30
which is the best one
GPT 3.5
The set of side lengths that best satisfies the Triangle Inequality Theorem and can form a triangle is 10, 20, 15. Each pair of side lengths in this set satisfies the inequality condition.