To find the location of Mac's house, we need to first find the point that is 1/3 of the distance from Nate's house to the park. We can use the midpoint formula to do this:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
where (x1, y1) is the location of Nate's house and (x2, y2) is the location of the park.
Using the formula, we get:
Midpoint = [(-4 + 12)/2, (6 + 4)/2] = [4, 5]
This midpoint is exactly halfway between Nate's house and the park. To find the point that is 1/3 of the distance from Nate's house to the park, we need to go 1/3 of the distance from Nate's house to the midpoint.
The distance between Nate's house and the midpoint can be found using the distance formula:
Distance = √[(x2 - x1)² + (y2 - y1)²]
where (x1, y1) is the location of Nate's house and (x2, y2) is the midpoint.
Using the formula, we get:
Distance = √[(4 - (-4))² + (5 - 6)²] = √128
To find the point that is 1/3 of the distance from Nate's house to the midpoint, we can divide the distance by 3 and move that distance from Nate's house towards the midpoint.
The coordinates of the point that is 1/3 of the distance from Nate's house to the midpoint are:
[(4 - (-4))/3, (5 - 6)/3] = [8/3, -1/3]
To find the location of Mac's house, we need to go 2/3 of the distance from the midpoint to the park, in the opposite direction. The coordinates of Mac's house are:
[(4 + 2/3), (5 - 2/3)] = [22/3, 14/3]
Therefore, the location of Mac's house is (22/3, 14/3).