Which three lengths could be the lengths of the sides of a triangle?
A. 6 cm, 10cm, 9cm
B. 8cm, 4cm, 2cm
C. 7cm, 7cm, 34cm
D. 10cm, 25cm, 15cm
The three lengths that could be the lengths of the sides of a triangle are:
A. 6 cm, 10 cm, 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, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's evaluate each option:
A. 6 cm, 10 cm, 9 cm
Sum of the two shorter sides: 6 cm + 9 cm = 15 cm
Is the sum greater than the longest side? 15 cm > 10 cm
Answer: A could be the lengths of the sides of a triangle.
B. 8 cm, 4 cm, 2 cm
Sum of the two shorter sides: 4 cm + 2 cm = 6 cm
Is the sum greater than the longest side? 6 cm > 8 cm
Answer: B could NOT be the lengths of the sides of a triangle.
C. 7 cm, 7 cm, 34 cm
Sum of the two shorter sides: 7 cm + 7 cm = 14 cm
Is the sum greater than the longest side? 14 cm > 34 cm
Answer: C could NOT be the lengths of the sides of a triangle.
D. 10 cm, 25 cm, 15 cm
Sum of the two shorter sides: 10 cm + 15 cm = 25 cm
Is the sum greater than the longest side? 25 cm > 25 cm
Answer: D could be the lengths of the sides of a triangle.
Based on the triangle inequality theorem, the correct answers are A and D. So, the three lengths that could be the lengths of the sides of a triangle are 6 cm, 10 cm, 9 cm (A) and 10 cm, 25 cm, 15 cm (D).
To determine if three lengths could be the lengths of the sides of a triangle, we need to apply the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let's evaluate each option:
A. 6 cm, 10 cm, 9 cm
In this case, 6 + 9 = 15, which is greater than 10. Also, 6 + 10 = 16, which is greater than 9. Finally, 9 + 10 = 19, which is greater than 6. Therefore, this set of lengths could be the lengths of the sides of a triangle.
B. 8 cm, 4 cm, 2 cm
Here, 8 + 4 = 12, which is greater than 2. Also, 8 + 2 = 10, which is greater than 4. However, 4 + 2 = 6, which is not greater than 8. Therefore, this set of lengths could NOT be the lengths of the sides of a triangle.
C. 7 cm, 7 cm, 34 cm
In this case, 7 + 7 = 14, which is greater than 34. Additionally, 7 + 34 = 41, which is greater than 7. However, 7 + 34 = 41, which is NOT greater than 7. Hence, this set of lengths could NOT be the lengths of the sides of a triangle.
D. 10 cm, 25 cm, 15 cm
Here, 10 + 25 = 35, which is greater than 15. Also, 10 + 15 = 25, which is greater than 25. Finally, 25 + 15 = 40, which is greater than 10. Therefore, this set of lengths could be the lengths of the sides of a triangle.
Based on our evaluations, the three lengths that could be the lengths of the sides of a triangle are option A: 6 cm, 10 cm, 9 cm, and option D: 10 cm, 25 cm, 15 cm.