The simplest figure in geometry is referred to as a point. It does not have a certain size and is often represent with a dot and labeled using capital letters. Points are in all geometric figures and we define space to be the set of all points. As such single points have no characteristics, when the points are grouped, we get different types of points coplanar, non-coplanar, collinear and non-collinear.

## Collinear Points

Points lying on one line are called as Collinear. Any two points are ever collinear because when connected, they determine a line.

Three or more points can be collinear but it is not necessary that all the points has to be collinear.
In the above figure P, Q, and R are collinear points.

In Cartesian plane any three points are collinear (say the points are A($x_1, y_1$) B($x_2, y_2$) and C($x_3, y_3$) if the ratios $\frac{(y_2 - y_1)}{(x_2 - x_1)}$ = $\frac{(y_3 – y_1)}{(x_3 – x_1)}$ = $\frac{(y_3 – y_2)}{(x_3 – x_2)}$ are equal.

In other words the slope of line formed by any two given points is same.

Example: Determine if the points A(2, 6), B(-1, 0) and C(5, 12) are collinear.
Solution:Let us determine slope formed by pair of points say A and B.

A(2, 6) B(-1, 0)
slope of line joining AB = $\frac{(0 - 6)}{(-1 - 2)}$

= $\frac{-6}{-3}$

Slope = 2

Check slope of line joining B and C.
B(-1, 0) and C(5, 12)
Slope of line joining B and C = $\frac{(12 - 0)}{(5-(-1)}$

= $\frac{12}{6}$

Slope = 2

Since slope of lines joining AB and BC are equal, the points A, B and C are collinear.

### Non-collinear points

These are the points which will not lie on the same plane, in the above figure X, Y, and Z are example for non-collinear points.

If three points are given say X Y and Z and slopes of lines formed by the pairs of points do not match then the points are not lying on the same line. Such points are called as non collinear.

Example: Determine if the points X(-2, -4), y(0, -1) and Z(3, 5)
Solution: To check if the three points are collinear let us determine the sope of lines formed by pair of points say X-Y and Y-Z.
Slope of line joining X and Y
X(-2, -4) and Y(0, -1)

slope = $\frac{(-1 - (-4))}{(0 - (-2))}$

= $\frac{3}{2}$

Determine the slope of line joining Y and Z

Y(0, -1) and Z(3, 5)

Slope = $\frac{(5 - (-1))}{(3 - 0)}$

= $\frac{6}{3}$ = 2

Since the slope of XY is not equal to the slope of YZ the points are not lying on the same line and hence are non collinear.

## Coplanar Points

A set of points, line portions, lines, rays or any other geometrical shapes that lie on the same plane are said to be Coplanar. Similar lines in three-dimensional space are coplanar. Four or more points might or might not be coplanar.

Left figure above shows coplanar points A, B, C, and D. At the right we have the cuboid drawn, there are many sets of coplanar points. For example points W, X, Q, and P are coplanar. Each of the six faces of the box contains four coplanar points, but these are not the only groups of coplanar points. The points Z, S, X, and Q are coplanar even though the plane that includes them isn’t shown; it slices the box in half diagonally.

### Non-coplanar points

These are the group of points that do not lie in the same plane.

In the above figure, points X, Y, P, and Q are non-coplanar. The top of the box contains Q, X, R and Y, and the left side contains P, Q, and X, but no flat surface contains all four points.

Example: Give two sets of each coplanar and non coplanar points from the given diagram.

Set of Coplanar points.
Set 1: {P, T, V, S}
Set 2: {P, Q, R, S}
Set of non coplanar points
Set 1: {P, Q, W, T}
Set 2: {R, U, W, P}

## Intersecting Points

Intersection of two figures is the set of all points that are common to both the figures.
Two lines intersect at a point. Two lines can intersect each other at one and only one point.This property is applied while solving linear system of equations graphically.
Two planes intersect at a common line.

Let line l and m intersect at point A .

The point of intersection can be determined either by algebraic method or by graphing.

Since the point of intersection lies on both the lines it satisfies equations of both the lines.

Example: Determine if the lines l, y = 2x - 5 and m: y = -x + 4 intersect. If yes, then determine the point of intersection.

Solution:
Let us first determine algebraically.

We have two linear equations which can be solved by using elimination –substitution method.
y = 2x - 5 ……..(1)
y = -x + 4……..(2).

Since the coefficient of y is 1 (same) we can use substitution.
Plug in y = 2x - 5 in equation (2).

2x - 5 = -x + 4

Solve for x

2x - 5 + x = -x + 4 + x

3x - 5 = 4

3x - 5 + 5 = 4 + 5
3x = 9

divide both sides by 3

$\frac{3x}{3}$ = $\frac{9}{3}$

so x = 3

Now to get the ordered pair plug in x=3 in any of the equation.
So equation (2) becomes
y = -3 + 4
y = 1

So the point of intersection is (3, 1).

Let us graph the lines and check their point of intersection.

Use point chart to draw the lines.
y = -x + 4
x y
04
13
22

y = 2x - 5
x y
0-5
1-3
2-1

From the graph we can see that the point of intersection is (1, 3) same as obtained from algebraical method.