How many triangles can you make with a perimeter of 12 units
many. the only restriction of the legs is that any one of them cannot be equal nor greater than the sum of the other two.
2,2,8 not allowed, 8 is greater then sum 2+2
1.7,6,4.3 not allowed, six is equal to the sum
1.9,5,5.1 is allowed. No side is greater nor equal to the sum of the other two sides
To determine how many triangles can be formed with a perimeter of 12 units, we can use the Triangle Inequality Theorem. According to this theorem, the sum of any two sides of a triangle must be greater than the length of the third side.
Let's assume the three sides of the triangle are labeled as a, b, and c. From the Triangle Inequality Theorem, we can derive the following inequalities:
a + b > c
b + c > a
a + c > b
Considering the perimeter is the sum of the three sides (a + b + c = 12), we can use the above inequalities to simplify the problem.
Since a, b, and c must be positive integers and their sum is 12, we can start by assigning the value '1' to one side and use trial and error to find the possible combinations:
- If we assign 1 to side a, then b + c = 11. The possible values for b and c could be (2, 9), (3, 8), (4, 7), (5, 6), or any other combination that sums up to 11.
- If we assign 1 to side b, then a + c = 11. Again, we can find various combinations like (2, 9), (3, 8), (4, 7), (5, 6), or others.
- Assigning 1 to side c would yield a + b = 11, leading to the same combinations as before.
We need to ensure that the found combinations satisfy the Triangle Inequality Theorem. For example, if a = 1, b = 4, and c = 7, we can check these values against the inequalities:
a + b > c : 1 + 4 > 7 (TRUE)
b + c > a : 4 + 7 > 1 (TRUE)
a + c > b : 1 + 7 > 4 (TRUE)
Therefore, a triangle with side lengths 1, 4, and 7 units can indeed be formed.
Repeating this process for all possible combinations, we can determine the number of triangles with a perimeter of 12 units.