Which of the following possibilities will form a triangle?
A. Side = [tex]$15 cm$[/tex], side = [tex]$6 cm$[/tex], side = [tex]$8 cm$[/tex]
B. Side = [tex]$15 cm$[/tex], side = [tex]$6 cm$[/tex], side = [tex]$9 cm$[/tex]
C. Side = [tex]$16 cm$[/tex], side = [tex]$9 cm$[/tex], side = [tex]$6 cm$[/tex]
D. Side = [tex]$16 cm$[/tex], side = [tex]$9 cm$[/tex], side = [tex]$8 cm$[/tex]
Answer (2)
Check if the sum of any two sides is greater than the third side for each option.
For sides 15, 6, and 8: 8"> 15 + 6 > 8 , 6"> 15 + 8 > 6 , but 6 + 8 < 15 , so it doesn't form a triangle.
For sides 15, 6, and 9: 9"> 15 + 6 > 9 , 6"> 15 + 9 > 6 , but 6 + 9 = 15 , so it doesn't form a triangle.
For sides 16, 9, and 6: 6"> 16 + 9 > 6 , 9"> 16 + 6 > 9 , but 9 + 6 < 16 , so it doesn't form a triangle.
For sides 16, 9, and 8: 8"> 16 + 9 > 8 , 9"> 16 + 8 > 9 , and 16"> 9 + 8 > 16 , so it forms a triangle.
The only possibility that forms a triangle is 16 c m , 9 c m , 8 c m .
Explanation
Problem Analysis and Strategy To determine which set of side lengths can form a triangle, we need to apply the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We will check each option to see if it satisfies this condition.
Checking Option 1: 15, 6, 8 Let's analyze the first option: Side = 15 cm, side = 6 cm, side = 8 cm. We need to check the following inequalities:
8"> 15 + 6 > 8 (which simplifies to 8"> 21 > 8 , which is true)
6"> 15 + 8 > 6 (which simplifies to 6"> 23 > 6 , which is true)
15"> 6 + 8 > 15 (which simplifies to 15"> 14 > 15 , which is false)
Since the third inequality is false, this set of side lengths cannot form a triangle.
Checking Option 2: 15, 6, 9 Now let's analyze the second option: Side = 15 cm, side = 6 cm, side = 9 cm. We need to check the following inequalities:
9"> 15 + 6 > 9 (which simplifies to 9"> 21 > 9 , which is true)
6"> 15 + 9 > 6 (which simplifies to 6"> 24 > 6 , which is true)
15"> 6 + 9 > 15 (which simplifies to 15"> 15 > 15 , which is false)
Since the third inequality is false, this set of side lengths cannot form a triangle.
Checking Option 3: 16, 9, 6 Next, let's analyze the third option: Side = 16 cm, side = 9 cm, side = 6 cm. We need to check the following inequalities:
6"> 16 + 9 > 6 (which simplifies to 6"> 25 > 6 , which is true)
9"> 16 + 6 > 9 (which simplifies to 9"> 22 > 9 , which is true)
16"> 9 + 6 > 16 (which simplifies to 16"> 15 > 16 , which is false)
Since the third inequality is false, this set of side lengths cannot form a triangle.
Checking Option 4: 16, 9, 8 Finally, let's analyze the fourth option: Side = 16 cm, side = 9 cm, side = 8 cm. We need to check the following inequalities:
8"> 16 + 9 > 8 (which simplifies to 8"> 25 > 8 , which is true)
9"> 16 + 8 > 9 (which simplifies to 9"> 24 > 9 , which is true)
16"> 9 + 8 > 16 (which simplifies to 16"> 17 > 16 , which is true)
Since all three inequalities are true, this set of side lengths can form a triangle.
Conclusion Therefore, the only set of side lengths that can form a triangle is Side = 16 cm, side = 9 cm, side = 8 cm.
Examples
The triangle inequality theorem is a fundamental concept in geometry and has practical applications in various fields. For example, in construction, when building a triangular structure like a roof truss, engineers must ensure that the lengths of the beams satisfy the triangle inequality to guarantee the structural integrity. If the inequality is not satisfied, the structure will not be stable and could collapse. Similarly, in navigation, understanding the triangle inequality helps in determining the shortest path between two points, especially when dealing with obstacles or indirect routes.
To find which sets of side lengths can form a triangle, we used the triangle inequality theorem. After checking all the options, only Option D, with lengths 16 cm, 9 cm, and 8 cm, satisfies the conditions. Therefore, this is the correct choice for forming a triangle.
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