(Multiple Choice Worth 5 points) (05.02 LC) Which of the following possibilities will form a triangle? Side = 16 cm, side = 8 cm, side = 7 cm Side = 16 cm, side = 9 cm, side = 7 cm Side = 17 cm, side = 12 cm, side = 7 cm Side = 17 cm, side = 11 cm, side = 5 cm Question 14
The Correct Answer and Explanation is:
To determine which set of sides can form a triangle, we 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. This must be true for all three combinations in the set.
Let’s evaluate each option:
Option 1: Sides = 16 cm, 8 cm, 7 cm
- 16 + 8 = 24 > 7 ✅
- 16 + 7 = 23 > 8 ✅
- 8 + 7 = 15 < 16 ❌
Not a triangle
Option 2: Sides = 16 cm, 9 cm, 7 cm
- 16 + 9 = 25 > 7 ✅
- 16 + 7 = 23 > 9 ✅
- 9 + 7 = 16 = 16 ❌
Equality is not enough. Must be strictly greater.
Not a triangle
Option 3: Sides = 17 cm, 12 cm, 7 cm
- 17 + 12 = 29 > 7 ✅
- 17 + 7 = 24 > 12 ✅
- 12 + 7 = 19 > 17 ✅
Valid triangle
Option 4: Sides = 17 cm, 11 cm, 5 cm
- 17 + 11 = 28 > 5 ✅
- 17 + 5 = 22 > 11 ✅
- 11 + 5 = 16 < 17 ❌
Not a triangle
✅ Correct answer: Option 3 — 17 cm, 12 cm, 7 cm
Explanation
To determine if three lengths can form a triangle, we use the Triangle Inequality Theorem. This rule is one of the foundational concepts in geometry, stating that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. The logic behind this rule is simple: if one side is too long compared to the other two, they will not be able to meet to form a closed figure, which is necessary for a triangle.
When applying this rule, we take each pair of sides and check whether their sum is greater than the third. This must hold for all three combinations of the side lengths. If even one of them fails the test, the side lengths cannot form a triangle.
For example, in option three, the side lengths are 17 cm, 12 cm, and 7 cm. Adding 17 and 12 gives 29, which is greater than 7. Adding 17 and 7 gives 24, which is greater than 12. Adding 12 and 7 gives 19, which is greater than 17. Since all three conditions are satisfied, these side lengths can form a triangle.
The other sets do not meet all three parts of the Triangle Inequality Theorem. In each failed case, at least one pair of sides adds up to less than or exactly equal to the third side. Equality is not enough, because the sides would lie flat and not enclose space, which is not a triangle. Therefore, only the set with sides 17 cm, 12 cm, and 7 cm forms a valid triangle.
