Distance from a point to a line (Coordinate Geometry)
Method 1: When the line is vertical or horizontal

Finding the distance from a point to a line is easy if the line is vertical or horizontal. We simply find the difference between the appropriate coordinates of the point and the line. In fact, for vertical lines, this is the only way to do it, since the other methods require the slope of the line, which is undefined for vertical lines. For more on this see Slope of a line (Coordinate Geometry)

Vertical lines

In the figure above, click on 'reset'. As you can see we have a vertical line whose equation is x=22. This means that all points of the line have an x-coordinate of 22. The given point C has coordinates of (42,7) which means it has a x-coordinate of 42. The distance between the point and line is therefore the difference between 22 and 42, or 20.

Example:

• In the figure above click on 'reset'.
• Drag the point C to left, past the y-axis, until is has the coordinates of (-10,15).
• The line has an x-coordinate of 22.
• The point C has a x-coordinate of -10.
• The distance from C to the line is therefore
  |-10-22 | = 32

Horizontal lines

In the figure above, click on 'reset', then 'horizontal'. As you can see we have a horizontal line whose equation is y=25. This means that all points of the line have a y-coordinate of 25. The given point C has coordinates of (39,7) which means it has a y-coordinate of 7. The distance between the point and line is therefore the difference between 25 and 7, or 18.

Example:

• In the figure above click on 'reset', then 'horizontal'
• Drag the point C to down, past the x-axis, until is has the coordinates (40,-6).
• The line has a y-coordinate of 25.
• The point C has a y-coordinate of -6.
• The distance from C to the line is therefore
  | -6-25 | = 31

Things to try

1. In the figure above, click 'reset', and 'hide details'
2. Drag the point C to any location and drag the orange dot on the line to any location.
3. Calculate the distance from the point to the line.
4. Click on 'show details' to see how you did.
5. Click on 'horizontal' and repeat.

Other methods

• Distance from a point to a line using line equations
• Distance from a point to a line using trigonometry
• Distance from a point to a line using a formula

Limitations

In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculatioons to be slightly off.

For more see Teaching Notes

Other Coordinate Geometry topics

• Introduction to coordinate geometry
• The coordinate plane
• The origin of the plane
• Axis definition
• Coordinates of a point
• Distance between two points

• Introduction to Lines
  in Coordinate Geometry
• Line (Coordinate Geometry)
• Ray (Coordinate Geometry)
• Segment (Coordinate Geometry)
• Midpoint Theorem

• Distance from a point to a line
• - When line is horizontal or vertical
• - Using two line equations
• - Using trigonometry
• - Using a formula

• Intersecting lines

• Cirumscribed rectangle (bounding box)
• Area of a triangle (formula method)
• Area of a triangle (box method)
• Centroid of a triangle
• Incenter of a triangle
• Area of a polygon
• Algorithm to find the area of a polygon
• Area of a polygon (calculator)
• Rectangle
• Definition and properties, diagonals
• Area and perimeter
• Square
• Definition and properties, diagonals
• Area and perimeter
• Trapezoid
• Definition and properties, altitude, median
• Area and perimeter
• Parallelogram
• Definition and properties, altitude, diagonals

• Print blank graph paper