Which three lengths could be the lengths of the sides of a triangle?
(1 point)
Responses
6 cm, 10 cm, 9 cm
6 cm, 10 cm, 9 cm
8 cm, 4 cm, 2 cm
8 cm, 4 cm, 2 cm
7 cm, 7 cm, 34 cm
7 cm, 7 cm, 34 cm
10 cm, 25 cm, 15 cm
10 cm, 25 cm, 15 cm
To determine whether three lengths could be the lengths of the sides of a triangle, we need to check if the sum of the two smaller lengths is greater than the longest length. Let's apply this rule to each set of lengths:
1) 6 cm, 10 cm, 9 cm:
The sum of the two smaller lengths is 6 cm + 9 cm = 15 cm, which is greater than the longest length of 10 cm. Therefore, these lengths could be the lengths of the sides of a triangle.
2) 8 cm, 4 cm, 2 cm:
The sum of the two smaller lengths is 2 cm + 4 cm = 6 cm, which is not greater than the longest length of 8 cm. Therefore, these lengths could not be the lengths of the sides of a triangle.
3) 7 cm, 7 cm, 34 cm:
The sum of the two smaller lengths is 7 cm + 7 cm = 14 cm, which is not greater than the longest length of 34 cm. Therefore, these lengths could not be the lengths of the sides of a triangle.
4) 10 cm, 25 cm, 15 cm:
The sum of the two smaller lengths is 10 cm + 15 cm = 25 cm, which is greater than the longest length of 25 cm. Therefore, these lengths could be the lengths of the sides of a triangle.
So, the three lengths that could be the lengths of the sides of a triangle are 6 cm, 10 cm, and 9 cm.
To determine which three lengths could be the lengths of the sides of a triangle, we need to apply the triangle inequality theorem. According to this theorem, for a triangle with side lengths a, b, and c:
- The sum of the lengths of any two sides of the triangle must be greater than the length of the third side.
- The difference between the lengths of any two sides of the triangle must be less than the length of the third side.
Let's go through each option:
1) 6 cm, 10 cm, 9 cm
To check if this is a valid triangle, we can compare the sum of two side lengths against the third side:
6 + 10 = 16 > 9 (satisfied)
6 + 9 = 15 > 10 (satisfied)
10 + 9 = 19 > 6 (satisfied)
Since both conditions are met, 6 cm, 10 cm, and 9 cm could be the lengths of the sides of a triangle.
2) 8 cm, 4 cm, 2 cm
Let's apply the triangle inequality theorem:
8 + 4 = 12 > 2 (satisfied)
8 + 2 = 10 > 4 (satisfied)
4 + 2 = 6 > 8 (not satisfied)
Since the third condition is not met, 8 cm, 4 cm, and 2 cm cannot be the lengths of the sides of a triangle.
3) 7 cm, 7 cm, 34 cm
Applying the theorem:
7 + 7 = 14 > 34 (not satisfied)
7 + 34 = 41 > 7 (satisfied)
7 + 34 = 41 > 7 (satisfied)
Since the first condition is not met, 7 cm, 7 cm, and 34 cm cannot be the lengths of the sides of a triangle.
4) 10 cm, 25 cm, 15 cm
Checking:
10 + 25 = 35 > 15 (satisfied)
25 + 15 = 40 > 10 (satisfied)
10 + 15 = 25 > 25 (not satisfied)
Since the third condition is not met, 10 cm, 25 cm, and 15 cm cannot be the lengths of the sides of a triangle.
Based on the analysis above, the lengths that could be the lengths of the sides of a triangle are 6 cm, 10 cm, and 9 cm.