How Do You Know if 2 Lines Intersect

Intersection of two straight lines (Coordinate Geometry)

The indicate of intersection of two non-parallel lines tin can exist institute from the
equations of the two lines.

Endeavour this Drag whatever of the 4 points beneath to motility the lines. Annotation where they intersect.

To find the intersection of ii directly lines:

1. First nosotros need the equations of the two lines. If you do non have the equations, meet Equation of a line - gradient/intercept form and Equation of a line - point/slope form (If one of the lines is vertical, encounter the section below).
2. And then, since at the betoken of intersection, the two equations will have the same values of x and y, we set up the two equations equal to each other. This gives an equation that we tin solve for x
3. We substitute that x value in one of the line equations (it doesn't matter which) and solve it for y.

This gives united states the x and y coordinates of the intersection.

Example

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

y = 3x-iii

y = ii.3x+4

At the point of intersection they will both have the aforementioned y-coordinate value, so we set the equations equal to each other:

3x-3 = 2.3x+iv

This gives us an equation in 1 unknown (x) which we tin can solve: Re-accommodate to go x terms on left

3x - 2.3x = 4+3

Combining like terms

0.7x = 7

Giving

x = x

To find y, just set up x equal to 10 in the equation of either line and solve for y: Equation for a line (Either line will practise)

y = 3x - 3

Set ten equal to 10

y = 30 - 3

Giving

y = 27

We now have both 10 and y, and so the intersection bespeak is (10, 27)

Which equation course to use?

Think that lines can exist described past the slope/intercept form and point/slope course of the equation. Finding the intersection works the same way for both. Just fix the equations equal equally in a higher place. For case, if y'all had two equations in indicate-gradient form:

y = 3(x-3) + 9

y = two.1(10+2) - 4

simply set them equal:

iii(x-three) + nine  =  2.1(x+2) - four

and proceed as higher up, solving for x, then substituting that value into either equation to find y.

The 2 equations need non even exist in the same form. Just set them equal to each other and proceed in the usual mode.

When one line is vertical

When ane of the lines is vertical, it has no defined gradient, so its equation volition look something like 10=12. See Vertical lines (Coordinate Geometry). Nosotros observe the intersection slightly differently. Suppose we have the lines whose equations are

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

On the vertical line, all points on it take an ten-coordinate of 12 (the definition of a vertical line), so we only ready ten equal to 12 in the outset equation and solve it for y.
Equation for a line:

y = 3x - 3

Gear up x equal to 12 Using the equation of the 2d (vertical) line

y = 36 - 3

Giving

y = 33

And so the intersection point is at (12,33).

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

When they are parallel

When two lines are parallel, they do not intersect anywhere. If you endeavor 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 y'all endeavor the in a higher place process you would write 3x+4 = 3x+eight. An obvious impossibility.

Segments and rays might non intersect at all

Fig 1. Segments do non intersect

In the example of two non-parallel lines, the intersection will e'er be on the lines somewhere. But in the case of line segments or rays which have a limited length, they might non actually intersect.

In Fig 1 nosotros come across two line segments that do non overlap and then have no indicate of intersection. However, if you apply the method above to them, you lot volition find the point where they would have intersected if extended plenty.

Things to endeavour

1. In the in a higher place diagram, press 'reset'.
2. Drag any of the points A,B,C,D around and note the location of the intersection of the lines.
3. Elevate a point to get two parallel lines and annotation that they have no intersection.
4. Click 'hide details' and 'show coordinates'. Move the points to whatever new location where the intersection is still visible. Summate 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 crusade calculatioons to be slightly off.

For more than run into Educational activity Notes

Other Coordinate Geometry topics

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

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

• Distance from a indicate 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
• Surface area of a polygon
• Algorithm to discover the area of a polygon
• Area of a polygon (figurer)
• Rectangle
• Definition and backdrop, diagonals
• Area and perimeter
• Foursquare
• Definition and backdrop, diagonals
• Area and perimeter
• Trapezoid
• Definition and backdrop, distance, median
• Area and perimeter
• Parallelogram
• Definition and properties, altitude, diagonals

• Print blank graph newspaper

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