Suppose you have 3 pieces of string measuring 4 inches, 5 inches, and 10 inches. How many unique triangles can you form with these pieces of string?
A. 0
B. 1
C. 2
D. Infinitely Many
Suppose I told you that in any triangle the sum of any two sides
must be greater than the third side.
0. It makes zero unique triangles.
To determine the number of unique triangles that can be formed from the given pieces of string, we need to apply the triangle inequality theorem. According to this theorem, for a triangle to be formed, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's consider each string length and see if it can form a triangle with the other two lengths:
1. The 4-inch string:
- Can form a triangle with the 5-inch string if (4 + 5) > 10. This is true since 4 + 5 = 9, which is less than 10.
- Cannot form a triangle with the 10-inch string since (4 + 10) is not greater than 5.
2. The 5-inch string:
- Can form a triangle with the 4-inch string if (5 + 4) > 10. This is true since 5 + 4 = 9, which is less than 10.
- Cannot form a triangle with the 10-inch string since (5 + 10) is not greater than 4.
3. The 10-inch string:
- Cannot form a triangle with either the 4-inch or 5-inch strings since (10 + 4) and (10 + 5) are both not greater than the remaining string length.
Based on the above analysis, only one unique triangle can be formed using the given pieces of string, which is formed by using the 4-inch and 5-inch strings. Therefore, the correct answer is B. 1.