Coordinate Geometry of straight Lines

The Distance Between Two Points

The distance between the points A(x1,Y1) and b(x2,Y2) is given by . The result is based on pythagoras theorem

The Midpoint of a Line Joining Two Points

The midpoint of the line joining the points (x1, y1) and (x2, y2) is:

[½(x1 + x2), ½(y1 + y2)]

Example

Find the coordinates of the midpoint of the line joining (1, 2) and (3, 1).

Midpoint = [½(3 + 1), ½(2 + 1)] = (2, 1.5)

The Gradient of a Line Joining Two Points

Parallel and Perpendicular Lines

If two lines are parallel, then they have the same gradient.

If two lines are perpendicular, then the product of the gradients of the two lines is -1.

Example

a) y = 2x + 1

b) y = -½ x + 2

c) ½y = x - 3

The gradients of the lines are 2, -½ and 2 in that order. Thus (a) and (b) and perpendicular, (b) and (c) are perpendicular and (a) and (c) are parallel.

The Equation of a Line Using One Point and the Gradient

The equation of a line which has gradient m and which passes through the point (x1, y1) is:

y - y1 = m(x - x1)

Example

Find the equation of the line with gradient 2 passing through (1, 4).

y - 4 = 2(x - 1)

y - 4 = 2x - 2

y = 2x + 2

Since m = y2 - y1

x2 - x1

The equation of a line passing through (x1, y1) and (x2, y2) can be written as:

y - y1 = y2 - y1

x - x1 x2 - x1

Coordinate Geometry: Straight Lines

Any straight line has an equation of the form

y=mx+c

where m, the gradient, is the height through which the line rises in one unit step in the horizontal direction, and c, the intercept, is the y-coordinate of the point of intersection between the line and the y-axis. This is shown in Figure a below.

Figure a: The straight line, y=mx+c

If we know the gradient m of a straight line with unknown intercept c, and the coordinates (x1 y1) of a point through which it passes, then we know that

y1 = mx1+c

and thus

c = y1−mx1 

If we substitute into

Y = mx+c

we get

y=mx−mx1+y1

which we can rearrange to give

y−y1 = m(x−x1)

So, for instance, the straight line through the point (3 1) with gradient 2 is given by

Y − 1 = 2(x−3) 

which gives

y = 2x−5 

If we know two points (x1 y1) and (x2 y2) through which passes a line with unknown gradient m and intercept c, then

y1 y2 = = mx1+c mx2+c

Subtracting the first equation from the second gives

Y 2−y1=m(x2−x1)

and thus

m = x2−x1y2−y1

The equation of the line is therefore

y−y1 = x2−x1y2−y1(x−x1) 

So, for instance, the straight line through (−1 −2) and (2 7) has equation

y+2=2+17+2(x+1) 

which gives

y = 3x+1

Intersection of two straight lines (Coordinate Geometry)

The point of intersection of two non-parallel lines can be found from the equations of the two lines.

To find the intersection of two straight lines:

1. First we require the equations of the two lines. If you do not have the equations, see Equation of a line - slope/intercept form and Equation of a line - point/slope form (If one of the lines is vertical, see the section below.

2. 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

3. We substitute that x value in one of the line equations (it doesn't matter which) and solve it for y.

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

Example

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

y = 3x-3

y = 2.3x+4

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

3x-3 = 2.3x+4

This offers us an equation in one unknown (x) which we can solve:

Re-arrange to obtain x terms on left 3x - 2.3x = 4+3

Joining like terms 0.7x = 7

Giving x = 10

To find y, simply set x equal to 10 in the equation of either line and solve for y:

Equation for a line y = 3x - 3 (Either line will do)

Set x equal to 10 y = 30 - 3 

Giving y = 27 

We at this point have both x and y, so the intersection point is (10, 27)

Which equation form to use?

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

y = 3(x-3) + 9

y = 2.1(x+2) - 4

Merely set them equal:

3(x-3) + 9 = 2.1(x+2) - 4

and move on as above, solving for x, then substituting that value into either equation to obtain y.

The two equations require not to even be in the same form. Just set them equal to each other and go ahead in the usual way.

When one line is vertical

When one of the lines is vertical, it has no defined slope, therefore its equation will look something like x=12. We obtain the intersection in a slightly different way. Assuming 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), therefore we 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 y = 36 - 3 From the equation of the second (vertical) line

Giving y = 33 

Therefore, 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 instance the lines y=3x+4 and y=3x+8 are parallel due to the fact that their slopes (3) are equal. If you try the above process you would write 3x+4 = 3x+8. A clear impossibility.

Segments and rays might not intersect at all

In the case of two non-parallel lines, the intersection will always be on the lines somewhere. But in the case of line segments or rays which have a limited length, they might not actually intersect.

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

Things to try

1. In the above diagram, in 3d form when you try to 'reset' it.

2. And drag any of the points A,B,C,D around and observe the location of the intersection of the lines.

3. Drag a point to obtain two parallel lines and observe that they have no intersection.

4. When you click 'hide details' and 'show coordinates'. Move the points to any fresh location where the intersection is still visible. Calculate the slopes of the lines and the point of intersection. Verify your result.

Limitations

In the interest of clearness, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculations to be a little bit off.

Distance between two points and the midpoint

The distance formula is an algebraic expression used to calculate the distance between two points with the coordinates (x1, y1) and (x2, y2).

The coordinates of a point are a pair of numbers that define its precise location on a two-dimensional plane. Bear in mind that the coordinate plane has two axes at right angles to each other, known as the x and y axis. The coordinates of a given point stands for how far along each axis the point is situated.

Ordered Pair

The coordinates are written as an "ordered pair". The letter P is simply the name of the point and is used to distinguish it from others.

The two numbers in parentheses are the x and y coordinate of the point. The first number (x) shows how far along the x (horizontal) axis the point is. The second is the y coordinate and shows how far up or down the y axis could go. It is known as an ordered pair due to the fact that the order of the two numbers matters - the first is constantly the x (horizontal) coordinate.

The sign of the coordinate is significant. A positive number means to go to the right (x) or up(y). Negative numbers mean to go left (x) or down (y)..

Abscissa

The abscissa is another name for the x (horizontal) coordinate of a point. Pronounced "ab-SISS-ah" (the 'c;' is silent). Not used very much. Most often, the term "x-coordinate" is used.

Ordinate

The ordinate is another name for the y (vertical) coordinate of a point. Pronounced "ORD-inet". Not used very much. Most often, the term "y-coordinate" is used.

The point A is in the top right quadrant (first quadrant). Observe the way both x and y coordinates are positive due to the fact that the point is up and to the right of the origin.

Drag the point into the top left quadrant (second quadrant). Observe now that the x-coordinate is negative due to the fact that it is to the left of the origin, where x values are negative.

Move the point to the lower right quadrant (fourth quadrant). The x-coordinate is positive again due to the fact that it is to the right of the origin, but now the y coordinate is negative, due to the fact that it is below the origin. View more on psalmfresh.blogspot.com for more educating and tricks article Thank you