Answer: The location is (8.88, -1.84)
Step-by-step explanation:
We know that the school is on the line:
x + 7*y = -4
or we can rewrite this as:
x = -4 - 7*y
Then, the points on this line can be written as:
(-4 - 7*y, y).
Then we can assume that the school is located at the point:
(-4 - 7*a, a)
and we need to find the value of a.
Now, we have two towns in the points:
A = (2, - 2)
B = (8, 5)
The distance between two points (a, b) and (c, d) is:
D = √( (a- c)^2 + (b - d)^2)
And we want that the school to be equidistant between the two cities, then if the distance to the citie A is:
Da = √( (2 + 4 + 7*a)^2 + (-2 - a)^2)
And the distance to the city B is:
Db = √( (8 + 4 + 7*a)^2 + (5 - a)^2)
Then we must have that:
Da = Db
√( (2 + 4 + 7*a)^2 + (-2 - a)^2) = √( (8 + 4 + 7*a)^2 + (5 - a)^2)
then:
(2 + 4 + 7*a)^2 + (-2 - a)^2 = (8 + 4 + 7*a)^2 + (5 - a)^2
Let's solve this for a.
(6 + 7*a)^2 + (-2 - a)^2 = (12 + 7*a)^2 + (5 - a)^2
36 + 84*a + 49*a^2 + 4 + 4*a + a^2 = 144 + 168*a + 49*a^2 + 25 - 10*a + a^2
let's simplify this:
36 + 84*a + 4 + 4*a = 144 + 168*a + 25 - 10*a
-129 = 70*a
-129/70 = a = -1.84
Then the location of the school is:
y = -1.84
x = -4 - 7*-1.84 = 8.88
(8.88, -1.84)
