Final answer:
The length of the fence needed to enclose the triangular lot can be found by calculating the distance between each pair of vertices using the distance formula and then summing them up.
Step-by-step explanation:
To find the length of the fence needed to enclose a triangular lot, we can use the distance formula to calculate the length of each side of the triangle. The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we can calculate the distance between points A and B, points B and C, and points C and A. Using the coordinates, we get:
dAB = sqrt((4 - 2)^2 + (9 - 3)^2) = sqrt(2^2 + 6^2) = sqrt(4 + 36) = sqrt(40)
dBC = sqrt((-2 - 4)^2 + (7 - 9)^2) = sqrt((-6)^2 + (-2)^2) = sqrt(36 + 4) = sqrt(40)
dCA = sqrt((2 - (-2))^2 + (3 - 7)^2) = sqrt(4^2 + (-4)^2) = sqrt(16 + 16) = sqrt(32)
Therefore, the length of the fence needed to enclose the triangular lot is dAB + dBC + dCA = sqrt(40) + sqrt(40) + sqrt(32).