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: Use the sharper form of the triangle inequality.
For any triangle, the longest side must be strictly less than the sum of the other two, and also strictly more than their difference. If a side equals the sum of the other two exactly, the triangle simply flattens into a straight line, it cannot close up.
Step 2: Test each option, paying attention to equality cases.
For 3, 4, 5: $3+4 = 7 \gt 5$, fine. For 2.5, 3.5, 4.5: $2.5+3.5=6 \gt 4.5$, fine. For 2.3, 6.4, 5.2: $2.3+5.2=7.5 \gt 6.4$, fine. For 2, 4, 6: \[ 2 + 4 = 6 \] which is equal to the third side, not greater than it.
Step 3: Conclude.
Since equality here means the three lengths just lie flat along a line instead of forming a closed triangle, this is the set that fails. \[ \boxed{2 \text{ cm}, 4 \text{ cm}, 6 \text{ cm}} \]