Among the following options, which are NOT sides of a triangle?
3 cm, 5 cm and 1 cm
Understanding Triangle Side Lengths
The question asks us to identify which set of given lengths cannot form the sides of a triangle. To determine if three lengths can form a triangle, we use a fundamental concept in geometry called the Triangle Inequality Theorem.
The Triangle Inequality Theorem
The Triangle Inequality 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. If this condition is not met for even one pair of sides, then the three lengths cannot form a triangle.
Let the three side lengths be $a$, $b$, and $c$. For these lengths to form a triangle, the following three inequalities must all be true:
- $a + b > c$
- $a + c > b$
- $b + c > a$
Applying the Triangle Inequality Theorem to the Options
Let's check each set of side lengths provided in the options against the Triangle Inequality Theorem.
| Option | Side Lengths (a, b, c) | Check 1: a + b > c | Check 2: a + c > b | Check 3: b + c > a | Can form a triangle? |
|---|---|---|---|---|---|
| 1 | 12 cm, 9 cm, 15 cm | $12 + 9 = 21 > 15$ (True) | $12 + 15 = 27 > 9$ (True) | $9 + 15 = 24 > 12$ (True) | Yes |
| 2 | 20 cm, 20 cm, 20 cm | $20 + 20 = 40 > 20$ (True) | $20 + 20 = 40 > 20$ (True) | $20 + 20 = 40 > 20$ (True) | Yes |
| 3 | 3 cm, 5 cm, 4 cm | $3 + 5 = 8 > 4$ (True) | $3 + 4 = 7 > 5$ (True) | $5 + 4 = 9 > 3$ (True) | Yes |
| 4 | 3 cm, 5 cm, 1 cm | $3 + 5 = 8 > 1$ (True) | $3 + 1 = 4$ < $5$ (False) | $5 + 1 = 6 > 3$ (True) | No |
Analysis of Options
- Option 1: 12 cm, 9 cm, and 15 cm satisfy all three conditions of the Triangle Inequality Theorem. So, these can be sides of a triangle.
- Option 2: 20 cm, 20 cm, and 20 cm satisfy all three conditions. This is an equilateral triangle. So, these can be sides of a triangle.
- Option 3: 3 cm, 5 cm, and 4 cm satisfy all three conditions. So, these can be sides of a triangle (this is a right-angled triangle by the Pythagorean theorem, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$).
- Option 4: 3 cm, 5 cm, and 1 cm fail one of the conditions. Specifically, $3 + 1 = 4$, which is NOT greater than the third side, 5 cm ($4 \ngtr 5$). This violates the Triangle Inequality Theorem. Therefore, these lengths cannot form a triangle.
Based on the analysis using the Triangle Inequality Theorem, the set of lengths that are NOT sides of a triangle is 3 cm, 5 cm, and 1 cm.
Revision Table: Key Concepts
| Concept | Explanation | Relevance to Question |
|---|---|---|
| Triangle | A polygon with three sides and three angles. | The question is about the sides of a triangle. |
| Side Lengths | The measures of the segments that form the sides of the triangle. | We are given sets of side lengths to test. |
| Triangle Inequality Theorem | The sum of any two side lengths must be greater than the third side length. | This is the rule used to check if side lengths can form a triangle. |
Additional Information: Degenerate Triangles
In some contexts, if the sum of two sides equals the third side (e.g., $a + b = c$), the lengths are said to form a "degenerate triangle". This is essentially a straight line segment where the three vertices lie on a single line. However, for a non-degenerate triangle (the standard type we usually refer to), the sum must be strictly greater ($a + b > c$). The question implies a standard triangle, so we require the strict inequality.
Understanding the Triangle Inequality Theorem is crucial for solving problems related to possible side lengths of a triangle.
Important Questions from Triangles
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