Final answer:
To find the area of the triangle formed by the lines, the lines need to be in the form Ax + By + C = 0. The vertices are needed to use determinants to find the triangle's area, which requires solving the equations pairwise to find the intersection points.
Step-by-step explanation:
To find the area of the triangle formed by the three given lines without solving for the vertices, we can use the concept of determinants and the area of a triangle given three lines in the form Ax + By + C = 0. However, the given lines need to be rewritten in this form first:
- 7x - 2y = 0
- 7x + 2y - 10 = 0
- 9x + y = 0
Moving the constants to the other side:
- 7x - 2y = 0 → 7x - 2y + 0 = 0
- 7x + 2y - 10 = 0 → 7x + 2y = 10
- 9x + y = 0 → 9x + y + 0 = 0
Once in this form, we can write a matrix with the coefficients A, B, and C from the equations and calculate the determinant of this matrix. The area of the triangle can then be found using the formula:
Area = |1/2 * determinant of the matrix formed by coefficients A, B, and C|
Unfortunately, we do not have a straightforward process to find the area without actually calculating the vertices since the area formula using determinants requires explicit coordinates of the vertices. You would need to solve the equations pairwise to find the vertices of the triangle first.