A house is to be connected to a new water main that runs alo

A house is to be connected to a new water main
that runs along the line y=1/2x-4. The connection point at the house has coordinates(1,4), where the units represent metres. What length of plastic pipe is needed to connect to the water main at the closest point? Solve this question algebraically, but a graph could be included, (HINT: the shortest distance between a point and a line is the perpendicular distance).

Solution

The pipeline has slope = 1/2
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Any line perpendicular to it must have slope -2
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Equation of the line perpendicular to the pipeline and passing thru (1,4)
Form: y = mx + b
You know y,x,and m; so solve for \"b\".
4 = -2*1 + b
b = 6
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EQUATION of the pipe connecting the house to the pipeline.
y = -2x + 6
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Find the point of intersection of the two pipes:
y = -2x + 6
y = (1/2)x - 4
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Sustitute to solve for \"x\":
-2x+6 = (1/2)x-4
(5/2)x = 10
x = 4
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Substitute to solve for \"y\":
y = -2*4+6 = -2
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Point of intersection is (4,-2)
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Find the distance from house to connection: (1,4) to (4,-2)
d = sqrt [(-2-4)^2 + (4-1)^2] = sqrt[36 + 9] = 3sqrt5 meters