How do you determine if a point lies on the left or right side of a line?

Question

using rectangular coordinates.

I mean a mathematical way to find the answer to my question.. not just by looking.

Answer 1

First check whether the origin is on the right or the left side of the line. Say that the origin is on the left side of the line.

Now substitute the origin in the line it gives a negative or positive number. Now substitute the given point in the line equation. Here if you get the same sign as when the origin has the the point lies to the left side and other wise to the right side.

For example line equation is 2x + 3y + 4 = 0
(0,0) give 4 which is positive.
Now check the given point's sign. Say (3,4)
This gives 22 which is positive and this implies that this point is on the same side of the origin. You can easily determine to which side is origin present.

This is how it is determined.

Hope you understand this.

Answer 2

Take the equation of the line and substitute the values of the point under consideration.
Example:
y = 2x + 4 Pt.(2,4)
4 = (2)2 + 4
4 ≈ 8 ( ≈ means not equal to).

Now, subtract the two numbers on the final line going from left to right.
4 - 8 = - 4 The sign is minus → The point is to the left.


When the sign is minus → This point is below the line or to the left.
When the sign is positive → The point is above the line or to the right.

Answer 3

If the slope of the line is positive, then it is to the left if it is above the line and to the right of the line if it is below the line. If the slope is negative, then switch.

If the line is vertical, then the point is to the right if the X coordinate of the point is greater than the X coordinate of all the points on the line and left if is is less.

If the slop is zero, then the point will not be left or right.

This isn't particularly important mathematics.

Answer 4

Sit down at a table with a blank sheet of paper. Draw everything I discuss below on that piece of paper.

Let's say you know two points on the line. Call them A and B. Now you want to know if X is to the left or to the right of line AB. Let's assume X is not on line AB. A line and a point-not-on-that-line define a plane. So you have a plane XAB.

Now you probably already have an xy rectangular coordinate system that lies in plane XAB. Forget about everything except what's in the plane XAB. This makes the whole problem just two-dimensional. Everything fits onto your paper at the kitchen table! Now you can find the answer.

Everything lies in the xy coordinate system (on your table). You know the xy coordinates of A and B.

Now pick a direction--either "from A to B" or "from B to A". Doesn't matter which one--just a decision to decide which direction is Left and which direction is Right. Get it?

For this discussion, let's say you picked "from A to B".

Now you are going to make a brand new rectangular coordinate system x'y' (x-prime, y-prime). Or you can call it rs coordinate system if you want--doesn't matter what you call it. The x'y' coordinate system also lies in plane XAB. The x'y' coordinate system is just the xy coordinate system moved h along the x-axis and k along the y-axis system until its origin lies at point A. The direction of increasing x' is in the same direction as x.

Find the equations for a coordinate transformation called an "origin translation". Now you can write the x'y' coordinates of point X in terms of its xy coordinates.

Now you are going to define a new coordinate system x''y'' whose origin is still at point A. The only difference between x'y' and x''y'' is that the x axis is rotated through an angle in the plane until it aligns with point B--in other words, the x'' axis is in the direction AB.

Find the equations for a coordinate transformation called an "origin rotation". Now you can write the x''y'' coordinates of point X in terms of its x'y'coordinates.

Its x'' coordinate lies along the ray AB (along A towards B), assuming I originally defined my direction as "from A to B".
Its y'' coordinate is its distance perpendicular to this line. If X's y'' coordinate is positive, X lies above the x'' axis, to the left of ray AB (or line AB from A towards B). If, on the other hand, X's y'' coordinate is negative, X lies below the x'' axis, to the right of ray AB. If X's y'' coordinate is zero, you probably already know X lies on line AB.

Answer 5

Look at the line, if the point is more (positive) than the points on the line, it's to the right I think. If it's less (negative) it's too the left. I'm not sure.

Answer 6

If you have the equation of the line,
Plug in the "y" value of the point given into the equation, and solve to find the corresponding X value on the line.

Now if the X value of the point is greater than the X value of the line, then the point is on the right and vice versa.

Answer 7

If the point is positive in x it is to the right and if the x of a point is - it is to the left.

eg; the point (3,2) lies to the right of the co-ordinate or the origin
the point (-3,2) lies to the left of the co-ordinate or the origin

Answer 8

I could easily tell by visually looking.. and I happen to know the difference between the word LEFT and the word RIGHT

Answer 9

Okay, this is tricky, but choose a side of the line to stand on, then say aloud, "This here towns not big enough for the two of us, when you cross this line, WE DUEL!"

Answer 10

well depending on you view to it as well as its X coordinates....if it's positive then right, negative then.....right! (I mean left!)