The number of triangles with any three of the lengths 1, 4, 6, and 8 cm is

A). One

B). Two

C). Three

D). Four

To determine how many triangles can be formed using any three lengths from the set {1 cm, 4 cm, 6 cm, 8 cm}, we need to apply the triangle inequality theorem. This theorem states that for any three sides of a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let's analyze the combinations step by step.

Identifying Combinations

First, we will list all the possible combinations of three lengths from the given set:

• 1 cm, 4 cm, 6 cm
• 1 cm, 4 cm, 8 cm
• 1 cm, 6 cm, 8 cm
• 4 cm, 6 cm, 8 cm

Applying the Triangle Inequality

Now, we will check each combination to see if it satisfies the triangle inequality conditions:

Combination: 1 cm, 4 cm, 6 cm

Check the inequalities:

• 1 + 4 > 6 (False)
• 1 + 6 > 4 (True)
• 4 + 6 > 1 (True)

This combination does not form a triangle because the first condition fails.

Combination: 1 cm, 4 cm, 8 cm

Check the inequalities:

• 1 + 4 > 8 (False)
• 1 + 8 > 4 (True)
• 4 + 8 > 1 (True)

This combination also does not form a triangle because the first condition fails.

Combination: 1 cm, 6 cm, 8 cm

Check the inequalities:

• 1 + 6 > 8 (False)
• 1 + 8 > 6 (True)
• 6 + 8 > 1 (True)

This combination does not form a triangle because the first condition fails.

Combination: 4 cm, 6 cm, 8 cm

Check the inequalities:

• 4 + 6 > 8 (True)
• 4 + 8 > 6 (True)
• 6 + 8 > 4 (True)

This combination satisfies all the conditions and can form a triangle.

Final Count of Triangles

After evaluating all combinations, we find that only one combination, {4 cm, 6 cm, 8 cm}, can form a triangle. Therefore, the answer to the question is:

A) One