Intersecting lines. (Coordinate Geometry) - Math Open Reference (2024)

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The point of intersectionof two non-parallellines can be found from the
equations of the two lines.

Try thisDrag any of the 4 points below to move the lines. Note where they intersect.

To find the intersection of two straight lines:

First we need the equations of the two lines. If you do not have the equations, seeEquation of a line - slope/intercept formand Equation of a line - point/slope form(If one of the lines is vertical, see the section below).
Then, since at the point of intersection, the two equations will have the same values of x and y, we set the two equations equal to each other.This gives an equation that we can solve for x
We substitute that x value in one of the line equations (it doesn't matter which) and solve it for y.

This gives us the x and y coordinates of the intersection.

Example

So for example, if we have two lines that have the following equations (in slope-intercept form):

y = 3x-3

Which equation form to use?

Recall that lines can be described by theslope/intercept formand point/slope formof the equation. Finding the intersection works the same way for both.Just set the equations equal as above. For example, if you had two equations in point-slope form:

y = 3(x-3) + 9

When one line is vertical

When one of the lines is vertical, it has no defined slope, so its equation will look something like x=12.See Vertical lines (Coordinate Geometry). We find the intersection slightly differently. Suppose we have the lines whose equations are

y = 3x-3	A line sloping up and to the right
x = 12	A vertical line

On the vertical line, all points on it have an x-coordinate of 12 (the definition of a vertical line), sowe simply set x equal to 12 in the first equation and solve it for y.
Equation for a line:

y = 3x - 3

Set x equal to 12 Using the equation of the second (vertical) line

y = 36 - 3

Giving

y = 33

So the intersection point is at (12,33).

If both lines are vertical, they are parallel and have no intersection (see below).

When they are parallel

When two lines are parallel, they do not intersect anywhere. If you try to find the intersection, the equations will be an absurdity.For example the lines y=3x+4 and y=3x+8 are parallel because their slopes (3) are equal. See Parallel Lines (Coordinate Geometry). If you try the above process you would write3x+4 = 3x+8. An obvious impossibility.

Segments and rays might not intersect at all

Fig 1. Segments do not intersect

In the case of two non-parallel lines, the intersection will always be on the lines somewhere. But in the case ofline segmentsor rayswhich have a limited length, they might not actually intersect.

In Fig 1 we see two line segments thatdo not overlap and so have no point of intersection. However, if you apply the method above to them, you will find the point where they would have intersected if extended enough.

Things to try

In the above diagram, press 'reset'.
Drag any of the points A,B,C,D around and note the location of the intersection of the lines.
Drag a point to get two parallel lines and note that they have no intersection.
Click 'hide details' and 'show coordinates'. Move the points to any new location where the intersection is still visible.Calculate the slopes of the lines and the point of intersection. Click 'show details' to verify your result.

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 seeTeaching Notes