please help me I need to finish this so that I can go into grade 11. A new water tower must be equidistant from three communities. They are located at (-25, 20) (-5, -40) and (15, 30). Find the coordinates of the point where the water tower should be built which i am assuming is the circumcenter. Please give all steps

I will describe the method for you, but you must do the work.

Label the points A, B, and C
The centre of the circle will lie on the intersection of the right bisectors of AB and AC

1. Find the midpoint of AB
2. find the slope of AB
3. The slope of the right-bisector will be the negative reciprocal of the slope of AB
e.g. slope AB = (20+40)/(-25 + 5) = 60/-20 = -3
so the slope of the right-bisector is 1/3
You now have the slope of that line and a point on it, (its midpoint)
Find the equation of the line

repeat the above steps to get the right-bisector of AC
solve the two equations, the solution would be the position of the tower.

hard

To find the coordinates of the point where the water tower should be built, which is the circumcenter of the triangle formed by the three communities, you can follow these steps:

Step 1: Calculate the midpoints of any two sides of the triangle.
Let's consider the midpoints of the sides formed by communities A and B, which we'll call AB_midpoint, and communities A and C, which we'll call AC_midpoint.

AB_midpoint = ((-25 + (-5)) / 2, (20 + (-40)) / 2)
= (-30 / 2, (-20 / 2))
= (-15, -10)

AC_midpoint = ((-25 + 15) / 2, (20 + 30) / 2)
= (-10 / 2, 50 / 2)
= (-5, 25)

Step 2: Calculate the slopes of the two sides of the triangle.
Using the coordinates of the three communities, we can calculate the slopes of the line segments AB and AC.

Slope_AB = (y2 - y1) / (x2 - x1)
= (-40 - 20) / (-5 - (-25))
= (-60) / (20)
= -3

Slope_AC = (y2 - y1) / (x2 - x1)
= (30 - 20) / (15 - (-25))
= (10) / (40)
= 1/4

Step 3: Calculate the perpendicular bisector slopes of the two sides.
The perpendicular bisector slopes are negative reciprocals of the slopes of the sides, denoted as m1 and m2.

m1 = -1 / Slope_AB
= -1 / -3
= 1/3

m2 = -1 / Slope_AC
= -1 / (1/4)
= -4

Step 4: Determine the equations of the perpendicular bisectors.
The equations of the perpendicular bisectors can be written in point-slope form using the midpoint and the corresponding slope.

Equation of AB_perpendicular_bisector:
y - y1 = m1 * (x - x1)
y - (-10) = (1/3) * (x - (-15))
y + 10 = (1/3)(x + 15)

Equation of AC_perpendicular_bisector:
y - y1 = m2 * (x - x1)
y - 25 = -4 * (x - (-5))
y - 25 = -4(x + 5)

Step 5: Solve the two equations to find the point of intersection.
Set the equations of the perpendicular bisectors equal to each other and solve for x and y.

(1/3)(x + 15) = -4(x + 5)

Multiply both sides by 3 to eliminate the fraction:
x + 15 = -12(x + 5)

Distribute the -12 on the right side:
x + 15 = -12x - 60

Combine like terms:
x + 12x = -60 - 15
13x = -75

Divide by 13:
x = -75 / 13

Substitute x into one of the perpendicular bisectors equations:
y = -4(x + 5) + 25
y = -4(-75/13 + 5) + 25
y = -4(-75/13 + 65/13) + 25
y = -4 * (-10/13) + 25
y = 40/13 + 25
y = 40/13 + (25*13)/13
y = 40/13 + 325/13
y = 365/13

Step 6: Write the coordinates of the point of intersection.
The coordinates of the point where the water tower should be built (circumcenter) are the values of x and y obtained from the previous step.

The coordinates are:
x = -75/13
y = 365/13

Therefore, the water tower should be built at the point (-75/13, 365/13).