maths geometry

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A straight road separates towns A (grid ref 42,45) and B (grid ref 52,40) A village is situated at grid ref (46,57) and a new road is planned from the village to road AB. What is the shortest road possible (1 dp)
Please define an origin and a set of axes
Use Cartesian geometry principles to work out lenght of the shortest line

The shortest route to the road would be perpendicular to the road.

The origin of coordinates has already been designated as where the grid is (0,0). The axis have also been defined as the map grid.

The "slope" of the road on the map from A to B, relative to the x axis, is
(40-45)/(52-42) = -1/2.

That means the slope (direction) of the line from the village at (46,57) to road AB is +2 (since it is perpendicular to AB).
The equation for the new road is
y - 57 = 2(x - 46)
y = 2x - 35
The roads connect where the equation of the AB road is also valid:
(y - 45) = (-1/2) (x - 42)
y = -(1/2)x + 66
Solve the two equations:
0 = 2.5 x - 101
x = 40.4
y = 45.8
Shortest new road length =
sqrt[(46 - 40.4)^2 + (57 - 45.8)^2]
= 12.52
Check the math; the method should be OK. Also try plotting it on graph paper to see if it makes sense.

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