Question

Which of the following cannot be the sides of a triangle ?

• A. 3 cm, 4 cm, 5 cm
• B. 2 cm, 4 cm, 6 cm
• C. 2.5 cm, 3.5 cm, 4.5 cm
• D. 2.3 cm, 6.4 cm, 5.2 cm

Hint:

To quickly test triangle validity, find the two smallest numbers in the set and add them together.
If their sum is less than or equal to the largest number, a triangle cannot be formed.
For \(2, 4, 6\): \(2 + 4 = 6\), which is equal to 6, failing the inequality immediately.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from Geometry, specifically concerning the Triangle Inequality Theorem.
We are given four sets of side lengths and need to determine which set cannot form a valid closed triangle.

Step 2: Key Formula or Approach:
The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining third side.
If \(a\), \(b\), and \(c\) are the side lengths (with \(c\) being the longest side), the triangle is valid if and only if:
\[ a + b \gt c \]

Step 3: Detailed Explanation:
Let us test each set of sides:
- Option (A): 3 cm, 4 cm, 5 cm
The longest side is 5. Sum of the two shorter sides:
\[ 3 + 4 = 7 \]
Since \(7 \gt 5\), these sides can form a triangle (specifically a right-angled triangle).
- Option (B): 2 cm, 4 cm, 6 cm
The longest side is 6. Sum of the two shorter sides:
\[ 2 + 4 = 6 \]
Since \(6\) is not strictly greater than \(6\) (\(6 = 6\)), these sides cannot form a triangle. They would form a straight line segment.
- Option (C): 2.5 cm, 3.5 cm, 4.5 cm
The longest side is 4.5. Sum of the two shorter sides:
\[ 2.5 + 3.5 = 6.0 \]
Since \(6.0 \gt 4.5\), these sides can form a triangle.
- Option (D): 2.3 cm, 6.4 cm, 5.2 cm
The longest side is 6.4. Sum of the two shorter sides:
\[ 2.3 + 5.2 = 7.5 \]
Since \(7.5 \gt 6.4\), these sides can form a triangle.

Step 4: Final Answer:
The set of sides that cannot form a triangle is 2 cm, 4 cm, 6 cm.