Distance from a point to a line (Coordinate Geometry)
Method 3: Using Trigonometry

NOTE: This method does not work if the line is horizontal or vertical (where the slope is undefined). In these cases use the the method described here.

In the figure above, we have a given line with the equation that describes it, and a point C with known coordinates. We want the perpendicular distance from C to the line at D. To find the distance CD:

1. Draw a horizontal line segment from C until it intersects the line at E, forming the right triangle CDE.
2. Find the coordinates of E The y coordinate of E must be the same as C, and the x coordinate is given by substituting that y into the line equation and solving for x.
3. Find the length of CE. By subtracting the x-coordinates of C and E we find the length of the line segment CE.
4. Find the angle E. This angle is the slope of the line in degrees. (See Slope of a line.)

   angle = |arctan m|
   where:
   m	is the slope part of the equation y=mx+b
   arctan	is the trigonometry inverse tan function. See Trigonometry Overview It means "find the angle whose tan is m"
   | |	the vertical bars mean "absolute value" - make it positive even if it calculates to a negative.

5. Find the distance CD.
   
   We know E and CE so we can solve for CD - the distance from the point to the line.

Example

1. Draw a horizontal line segment from C until it intersects the line at E, forming the right triangle CDE.
2. Find the coordinates of E The y coordinate of E must be the same as C which is 13, and the x coordinate is given by substituting y=13 into the line equation and solving for x: So E has the coordinates (15,13).
3. Find the length of CE. By subtracting the x-coordinates of C and E we find the length of the line segment CE to be 51.
4. Find the angle E. This angle is the slope of the line in degrees. (See Slope of a line.)
   angle E = |arctan 0.52| =27° (rounded to nearest degree)
5. Find the distance from the point C to the line (the length of CD).

Things to try

1. In the figure above, click 'reset', and 'hide details'
2. Drag the point C to any location and drag the two sliders to create a new line equation.
3. Calculate the distance from the point to the line.
4. Click on 'show details' to see how you did.

Other methods

• Distance from a point to a vertical or horizontal line
• Distance from a point to a line using line equations
• 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