How many different triangles can be formed having a perimeter of 7 units if each side must have integral length?
I got 2
2 2 3
3 3 1
I also got 2, like itrixie. I wonder if there are any scalene triangles though.
To determine the number of different triangles that can be formed with a perimeter of 7 units, we need to consider the possible side lengths for the triangle.
Since we are given that each side must have integral length, we can start by assuming the smallest possible side length, which is 1 unit. Let's call this side length "a".
Now, let's consider the other two side lengths. Since the perimeter of the triangle is given as 7 units, the sum of the two remaining side lengths must be 7 - 1 = 6 units. Let's call these two side lengths "b" and "c".
To determine the range of values for "b" and "c", we can use the following inequalities based on the triangle inequality theorem:
b + c > a (sum of any two sides must be greater than the third side)
b + a > c
c + a > b
Substituting the values, we have:
b + c > 1
b + 1 > c
c + 1 > b
Since b and c are both positive integers, the smallest possible value for each is 1. Therefore, we have:
1 + 1 > c
1 + 1 > b
Simplifying, we get:
2 > c
2 > b
Since c and b must both be positive integers, the possible values for c and b are 2, 3, 4... up to 6.
To find the number of different triangles, we can systematically count the possible combinations of b and c:
1. If b = 2, then c can be 4, 5, or 6.
2. If b = 3, then c can be 3, 4, or 5.
3. If b = 4, then c can be 2, 3, or 4.
4. If b = 5, then c can be 1, 2, or 3.
5. If b = 6, then c can only be 1.
Adding up the combinations, we get:
With a = 1, there are 3 different triangles.
With a = 2, there are 2 different triangles.
With a = 3, there is 1 different triangle.
Therefore, the total number of different triangles that can be formed with a perimeter of 7 units and integral side lengths is 3 + 2 + 1 = 6 triangles.