Try this
Drag any of the 4 points below to move the lines. Note they are parallel when the slopes are the same.

When two straight lines are plotted on the coordinate plane, we can tell if they are parallel from the slope, of each line. If the slopes are the same then the lines are parallel. In the figure above, there are two lines that are determined by given points. Drag any point to reposition the lines and note that they are parallel only when the slopes are equal.

The slope can be found using any method that is convenient to you:

- From two given points on the line. (See Slope of a line).
- From the equation of the line in slope-intercept form
- From the equation of the line in point-slope form

Fig 1. Are these lines parallel?

In Fig 1 there are two lines. One line is defined by two points at (5,5) and (25,15). The other is defined by an equation in slope-intercept form form y = 0.52x - 2.5. We are to decide if they are parallel.

For the top line, the slope is found using the coordinates of the two points that define the line. (See Slope of a Line for instructions).

For the lower line, the slope is taken directly from the formula. Recall that the slope intercept formula is
y = mx + b, where *m* is the slope. So looking at the formula we see that the slope is 0.52.

So, the top one has a slope of 0.5, the lower slope is 0.52, which are not equal. Therefore, the lines are __not parallel__.
The lines are *very close* to being parallel, and may look parallel, but appearance can deceive.

Fig 2. we need a line parallel to AB through C

We first find the slope of the line AB using the same method as in the example above.

For the line to be parallel to AB it will have the same slope, and will pass through a given point, C(12,10). We therefore have enough information to define the line by its equation in point-slope form form:

y = -0.52(x-12)+10

This is one of the ways a line can be defined and so we have solved the problem.
If we wanted to go ahead and actually plot the line
we can do so by finding another point on the line using the equation and then draw the line through the two points.
For more on this see Equation of a Line (point - slope form)
- In the above diagram, press 'reset'.
- Note that because the slopes are the same, the lines are parallel.
- Adjust one of the points defining the lines. They are no longer parallel.
- Drag a point on the other line to make them parallel again.
- Drag a point until the lines are not parallel. Click on "hide details". Determine the slope of both lines and prove they are not parallel. Click "show details" to verify.

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

- 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

- 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
- Square
- Trapezoid
- Parallelogram

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