A water tower has to be equidistant from three communities. The communities are at positions (-25, 20), (-5, -40) and (15, 30). Find the coordinates of the point where the water tower should be built.
To find the coordinates of the point where the water tower should be built, we need to find the intersection of the perpendicular bisectors of the three line segments connecting the communities.
Step 1: Calculate the midpoints of the line segments connecting the communities.
- The midpoint (M1) between (-25, 20) and (-5, -40) is:
M1 = ((-25 + (-5)) / 2, (20 + (-40)) / 2)
= (-30 / 2, -20 / 2)
= (-15, -10)
- The midpoint (M2) between (-25, 20) and (15, 30) is:
M2 = ((-25 + 15) / 2, (20 + 30) / 2)
= (-10 / 2, 50 / 2)
= (-5, 25)
- The midpoint (M3) between (-5, -40) and (15, 30) is:
M3 = ((-5 + 15) / 2, (-40 + 30) / 2)
= (10 / 2, -10 / 2)
= (5, -5)
Step 2: Calculate the slopes of the line segments connecting the communities.
- The slope (S1) between (-25, 20) and (-5, -40) is:
S1 = (y2 - y1) / (x2 - x1)
= (-40 - 20) / (-5 - (-25))
= (-60) / (20)
= -3
- The slope (S2) between (-25, 20) and (15, 30) is:
S2 = (y2 - y1) / (x2 - x1)
= (30 - 20) / (15 - (-25))
= (10) / (40)
= 1/4
- The slope (S3) between (-5, -40) and (15, 30) is:
S3 = (y2 - y1) / (x2 - x1)
= (30 - (-40)) / (15 - (-5))
= (70) / (20)
= 7/2
Step 3: Calculate the slopes of the perpendicular bisectors of the line segments.
- The slope (SP1) of the perpendicular bisector to the line segment connecting (-25, 20) and (-5, -40) is the negative reciprocal of S1, which is:
SP1 = -1 / S1
= -1 / (-3)
= 1/3
- The slope (SP2) of the perpendicular bisector to the line segment connecting (-25, 20) and (15, 30) is the negative reciprocal of S2, which is:
SP2 = -1 / S2
= -1 / (1/4)
= -4
- The slope (SP3) of the perpendicular bisector to the line segment connecting (-5, -40) and (15, 30) is the negative reciprocal of S3, which is:
SP3 = -1 / S3
= -1 / (7/2)
= -2/7
Step 4: Use the midpoints and slopes of the perpendicular bisectors to find the intersection point.
- For the perpendicular bisector passing through M1 (-15, -10) with slope SP1 = 1/3, the equation of the line is:
y - y1 = m(x - x1)
y - (-10) = (1/3)(x - (-15))
y + 10 = (1/3)(x + 15)
3y + 30 = x + 15
x - 3y - 15 = 0
- For the perpendicular bisector passing through M2 (-5, 25) with slope SP2 = -4, the equation of the line is:
y - y2 = m(x - x2)
y - 25 = (-4)(x - (-5))
y - 25 = (-4)(x + 5)
y - 25 = -4x - 20
4x + y - 45 = 0
- For the perpendicular bisector passing through M3 (5, -5) with slope SP3 = -2/7, the equation of the line is:
y - y3 = m(x - x3)
y - (-5) = (-2/7)(x - 5)
y + 5 = (-2/7)(x - 5)
7y + 35 = -2x + 10
2x + 7y - 25 = 0
Now we have three equations representing the perpendicular bisectors. We can solve them to find the intersection point, which will be the coordinates of the point where the water tower should be built.