George is making a triangular enclosure in his yard. He plans on using two old fence posts as vertices, and will place a new fence post for the third vertex. The old posts are located at (−5,4) and (2,6) , where the coordinates are given in feet. If he has 25 feet of wire fencing available to close off the area, which of the following are possible locations for the third fence post? Select the two correct answers.(1 point) Responses (−4,−5) open paren negative 4 comma negative 5 close paren (0,0) open paren 0 comma 0 close paren (−9,1) open paren negative 9 comma 1 close paren (2,−6) open paren 2 comma negative 6 close paren (8,8)
To find the possible locations for the third fence post, we can use the formula for the distance between two points in a coordinate plane.
The distance between the first fence post at (-5,4) and the second fence post at (2,6) is:
√((-5 - 2)^2 + (4 - 6)^2) = √((-7)^2 + (-2)^2) = √(49 + 4) = √53
Since the total amount of fencing wire available is 25 feet, the perimeter of the triangular enclosure must be less than or equal to 25 feet.
Consider the distance from the first fence post to each of the potential third fence posts:
1) Distance from (-5,4) to (-4,-5):
√((-5 - (-4))^2 + (4 - (-5))^2) = √((-1)^2 + (9)^2) = √(1 + 81) = √82
2) Distance from (-5,4) to (0,0):
√((-5 - 0)^2 + (4 - 0)^2) = √((-5)^2 + (4)^2) = √(25 + 16) = √41
3) Distance from (-5,4) to (-9,1):
√((-5 - (-9))^2 + (4 - 1)^2) = √((4)^2 + (3)^2) = √(16 + 9) = √25 = 5
4) Distance from (-5,4) to (2,-6):
√((-5 - 2)^2 + (4 - (-6))^2) = √((-7)^2 + (10)^2) = √(49 + 100) = √149
5) Distance from (-5,4) to (8,8):
√((-5 - 8)^2 + (4 - 8)^2) = √((-13)^2 + (4)^2) = √(169 + 16) = √185
Only when the distance from (-5,4) to (-9,1) or (0,0) is less than or equal to √53 and 25, is it possible to fence off the area with only 25 feet of wire fencing.
Therefore, the two possible locations for the third fence post are:
(−9,1)
(0,0)