How many different triangles with sides of measurements of 3, 4, and 5 are there?

one -- a right triangle

One

To determine the number of different triangles with side measurements of 3, 4, and 5, we can analyze the relationships between the side lengths.

In a triangle, the sum of any two sides must be greater than the length of the third side. If we apply this rule, we can evaluate the possibilities using the given side measurements.

In this case, the triangle with side lengths of 3, 4, and 5 satisfies this condition because:

3 + 4 > 5 (7 > 5)
4 + 5 > 3 (9 > 3)
3 + 5 > 4 (8 > 4)

Therefore, the triangle with side lengths of 3, 4, and 5 is valid.

However, there are no other different triangles possible with these side lengths because any other combination would violate the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

So, the answer is that there is only one different triangle with side lengths of 3, 4, and 5.