Understanding Triangle Sides and the Triangle Inequality Theorem

To determine if a set of three lengths can form the sides of a triangle, we use a fundamental principle in geometry called the Triangle Inequality Theorem. This theorem provides a condition that any valid set of triangle sides must satisfy.

The Triangle Inequality Theorem:

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the three side lengths be denoted by $a$, $b$, and $c$. For these lengths to form a triangle, the following three inequalities must all be true:

• $a + b > c$
• $a + c > b$
• $b + c > a$

If even one of these conditions is not met, then the given lengths cannot be the sides of a triangle.

Analyzing the Given Side Lengths

Let's apply the Triangle Inequality Theorem to each set of side lengths provided in the options to see which one cannot form a triangle.

Option 1: 3 cm, 4 cm, 5 cm

Let $a=3$, $b=4$, $c=5$. We check the three conditions:

• $a + b > c$: $3 + 4 = 7$. Is $7 > 5$? Yes.
• $a + c > b$: $3 + 5 = 8$. Is $8 > 4$? Yes.
• $b + c > a$: $4 + 5 = 9$. Is $9 > 3$? Yes.

Since all three conditions are met, 3 cm, 4 cm, and 5 cm can be the sides of a triangle. This is actually a right-angled triangle.

Option 2: 1 cm, 2 cm, 3 cm

Let $a=1$, $b=2$, $c=3$. We check the three conditions:

• $a + b > c$: $1 + 2 = 3$. Is $3 > 3$? No.
• $a + c > b$: $1 + 3 = 4$. Is $4 > 2$? Yes.
• $b + c > a$: $2 + 3 = 5$. Is $5 > 1$? Yes.

One of the conditions ($1 + 2 > 3$) is not met. The sum of the two shorter side lengths is equal to the longest side length, not greater than it. This means these side lengths cannot form a triangle; they would just form a straight line segment.

Option 3: 4 cm, 5 cm, 6 cm

Let $a=4$, $b=5$, $c=6$. We check the three conditions:

• $a + b > c$: $4 + 5 = 9$. Is $9 > 6$? Yes.
• $a + c > b$: $4 + 6 = 10$. Is $10 > 5$? Yes.
• $b + c > a$: $5 + 6 = 11$. Is $11 > 4$? Yes.

Since all three conditions are met, 4 cm, 5 cm, and 6 cm can be the sides of a triangle. These are valid triangle sides.

Option 4: 2 cm, 3 cm, 4 cm

Let $a=2$, $b=3$, $c=4$. We check the three conditions:

• $a + b > c$: $2 + 3 = 5$. Is $5 > 4$? Yes.
• $a + c > b$: $2 + 4 = 6$. Is $6 > 3$? Yes.
• $b + c > a$: $3 + 4 = 7$. Is $7 > 2$? Yes.

Since all three conditions are met, 2 cm, 3 cm, and 4 cm can be the sides of a triangle. These are also valid triangle sides.

Conclusion: Identifying Invalid Triangle Sides

Based on the analysis using the Triangle Inequality Theorem, the set of lengths that fails the condition is 1 cm, 2 cm, and 3 cm. Therefore, these cannot be the sides of a triangle. This shows how understanding triangle properties, particularly the triangle inequality theorem, helps determine valid side lengths in geometry problems involving triangle sides.