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.

To calculate the length of the shortest road from the village to the road AB, follow these steps:

1. Determine the slope of the road AB. The slope is calculated by taking the difference in y-coordinates and dividing it by the difference in x-coordinates between the two points A and B.
Slope = (y2 - y1) / (x2 - x1) = (40 - 45) / (52 - 42) = -1/2

2. The slope of the shortest road perpendicular to AB will be the negative reciprocal of the slope of AB. In this case, the perpendicular slope is 2.

3. Use the coordinates of the village (46,57) and the slope of the perpendicular road (2) to form the equation of the new road.
Equation: y - 57 = 2(x - 46)
Simplify to: y = 2x - 35

4. Find the point where the equation of the new road intersects with the equation of the road AB. Set the two equations equal to each other:
y - 45 = (-1/2)(x - 42)
Simplify to: y = -(1/2)x + 66

5. Solve the system of equations to find the x and y coordinates where the two roads intersect. Substitute the value of y from one equation into the other equation and solve for x:
2x - 35 = -(1/2)x + 66
Simplify and solve for x:
2.5 x = 101
x = 40.4

6. Substitute the value of x back into one of the equations to find the corresponding y-coordinate:
y = 2(40.4) - 35
y = 45.8

7. Calculate the length of the new road using the distance formula, considering the coordinates of the village (46,57) and the intersection point (40.4,45.8):
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
Length = √[(46 - 40.4)^2 + (57 - 45.8)^2]
Length = √[5.6^2 + 11.2^2]
Length = √[31.36 + 125.44]
Length ≈ 12.52 (rounded to 1 decimal place)

You can check the math and plot it on graph paper to visualize the result and verify the accuracy.