Question
Apply the triangle inequality theorem to determine which three side lengths form a triangle
The options are
10,20,15
8,4,12
20,10,30
8,8,20
The options are
10,20,15
8,4,12
20,10,30
8,8,20
Answers
GPT 3.5
To determine which three side lengths form a triangle, we need to apply the Triangle Inequality Theorem. According to this theorem, for a triangle to be formed, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's check each option:
1) 10, 20, 15:
10 + 20 = 30 (greater than 15)
20 + 15 = 35 (greater than 10)
15 + 10 = 25 (greater than 20)
Since all three sums are greater than the respective third side length, a triangle can be formed with side lengths 10, 20, and 15.
2) 8, 4, 12:
8 + 4 = 12 (equal to 12, the third side length)
4 + 12 = 16 (greater than 8)
12 + 8 = 20 (greater than 4)
In this case, the sum of the two shorter sides (8 and 4) is equal to the length of the longest side (12). Therefore, a triangle cannot be formed with side lengths 8, 4, and 12.
3) 20, 10, 30:
20 + 10 = 30 (equal to 30, the third side length)
10 + 30 = 40 (greater than 20)
30 + 20 = 50 (greater than 10)
Similar to the previous case, the sum of the two shorter sides (20 and 10) is equal to the length of the longest side (30). Therefore, a triangle cannot be formed with side lengths 20, 10, and 30.
4) 8, 8, 20:
8 + 8 = 16 (greater than 20)
8 + 20 = 28 (greater than 8)
20 + 8 = 28 (greater than 8)
In this case, the sum of the two shorter sides (8 and 8) is equal to the length of the longest side (20). Therefore, a triangle can be formed with side lengths 8, 8, and 20.
To summarize, the three side lengths that form a triangle are 10, 20, and 15.
Let's check each option:
1) 10, 20, 15:
10 + 20 = 30 (greater than 15)
20 + 15 = 35 (greater than 10)
15 + 10 = 25 (greater than 20)
Since all three sums are greater than the respective third side length, a triangle can be formed with side lengths 10, 20, and 15.
2) 8, 4, 12:
8 + 4 = 12 (equal to 12, the third side length)
4 + 12 = 16 (greater than 8)
12 + 8 = 20 (greater than 4)
In this case, the sum of the two shorter sides (8 and 4) is equal to the length of the longest side (12). Therefore, a triangle cannot be formed with side lengths 8, 4, and 12.
3) 20, 10, 30:
20 + 10 = 30 (equal to 30, the third side length)
10 + 30 = 40 (greater than 20)
30 + 20 = 50 (greater than 10)
Similar to the previous case, the sum of the two shorter sides (20 and 10) is equal to the length of the longest side (30). Therefore, a triangle cannot be formed with side lengths 20, 10, and 30.
4) 8, 8, 20:
8 + 8 = 16 (greater than 20)
8 + 20 = 28 (greater than 8)
20 + 8 = 28 (greater than 8)
In this case, the sum of the two shorter sides (8 and 8) is equal to the length of the longest side (20). Therefore, a triangle can be formed with side lengths 8, 8, and 20.
To summarize, the three side lengths that form a triangle are 10, 20, and 15.