Question 1038153: Madison is lost in the woods. She uses a GPS system to pinpoint herself at the point A (26, 18) on a coordinate grid of the area. She can locate a straight highway some distance away with gas stations at B (21, 2) and C (1, 22). What is the shortest distance to the highway from where Madison is?
*Analytic Geometry Question*
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! she is at (26,18)
highway has two points on it.
they are (21,2) and (1,22).
calculate straight line equation of the highway from the two points on it.
the slope intercept form of the equation for the straight line is y = mx + b, where m is the slope and b is the y-intercept.
the slope is equal to (y2-y1)/(x2-x1), where (x1,y1) and (x2,y2) are any two points on the line.
assign (1,22) to (x1,y1)
assign (21,2) to (x2,y2)
slope = (y2-y1)/(x2-x1) = (2-22)/(21-1) = -20/20 = -1.
formula becomes y = -x + b
to solve for b, assign any one of the points to x and y.
using (1,22), you get 22 = -1*1 + b
solve for b to get b = 23
the equation of the line of the highway is equal to y = -x + 23
the shortest distance from the point (26,18) to the line of the highway is a line perpendicular to it.
the slope of a line perpendicular to another line is the negative reciprocal of that line.
since the slope of the line of the highway is equal to -1, then the slope of a line perpendicular to it is 1.
the equation of the line perpendicular to the line of the highway is therefore y = x + b.
a point on that line is (26,18).
replace y with 18 and x with 26 to get 18 = 26*1 + b
simplify to get 18 = 26 + b
solve for b to get b = 18 - 26 = -8.
the equation of the line perpendicular to the line of the highway and passing through the point (26,18) is y = x - 8.
you have two equations that need to be solved simultaneously to find their intersection.
they are y = -x + 23 and y = x - 8
subtract the second equation from the first to get 0 = -2x + 31
add 2x to both sides of this equation to get 2x = 31.
solve for x to get x = 31/2 = 15.5
take the equation of y = x - 8 and replace x with 15.5 and solve for y to get y = 15.5 - 8 = 7.5
the point of intersection is (15.5,7.5)
the shortest distance from (26,18) to (15.5,7.5) is equal to sqrt((15.5-26)^2 + (7.5-18)^2 = sqrt(110.25 + 110.25) = sqrt(220.5) = 14.8492424 units.
assuming the distance is measured in miles, then the shortest distance to the highway would be 14.8492424 miles.
the graph of the equations of y = -x + 23 and y = x - 8 are shown below, along with the coordinate points of the problem.
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