Triangles are fascinating shapes. They have an array of properties and attributes that make them unique. One such feature is the area of a triangle, which is calculated using the base and height. Now, let’s consider a triangle GHJ. Given three sides with lengths of 5, 6, and 7 or 12, what could be the area of this triangle?

First, let’s deal with the case of 5, 6, and 7. We can use Heron’s formula to find the area of the triangle. Here, s is the semi-perimeter or half the perimeter of the triangle. Thus, s = (5 + 6 + 7)/2 = 9. Therefore, the area of the triangle is given by √(s(s-a)(s-b)(s-c)) = √(9(9-5)(9-6)(9-7)) = √(9 × 4 × 3 × 2) = 6√6.

Now, let’s consider the case where the sides have lengths of 12, 6, and 7. Since the sum of the smaller sides is less than the length of the longest side, we know that this triangle is not possible. Thus, this case is invalid, and we don’t need to calculate the area.

Therefore, the area of triangle GHJ with sides of 5, 6, and 7 is 6√6. It’s essential to note that knowing any three sides of a triangle determines the rest of its properties, such as its angles, area, perimeter, and more.

In conclusion, the area of the triangle GHJ can only be calculated when we have valid side lengths. Invalid side lengths do not offer a valid triangle, so it’s necessary to verify the triangle’s existence first.