Points, Lines and Planes Worksheet

  1. Identify which of the following sets of points are collinear: A(2, 3), B(5, 7), C(8, 11).
  2. Determine if the points D(4, 6), E(4, 6), and F(4, 8) lie on the same line.
  3. Given three non-collinear points, how many distinct lines can be drawn through them?
  4. Determine whether the points G(2, 3, 1), H(4, 1, 3), and I(6, 5, 2) lie on the same plane.
  5. Find the equation of the line passing through the points P(2, 1) and Q(4, 5) in slope-intercept form.
  6. Determine the distance between the points R(3, 2, 4) and S(7, 5, 8).

Points, Lines and Planes

Points, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using the coordinates.

We define a line as a geometrical figure that is extended in both directions to infinity. Similarly, a plane is defined as the collection of all such lines, i.e. it is a 3-D space on which the line passes.

In this article, we will learn about Points, Lines, and Planes in detail including their solved examples and problems based on them.

Table of Content

  • Points, Lines, and Planes in Geometry
  • What is a Point?
    • Collinear and Non-Collinear Points
    • Coplanar and Non-Coplanar Points
  • What is a Line?
    • Line Segment
  • Mid-Point
  • Rays
  • Intersecting and Parallel lines
  • Perpendicular Lines
  • What is a Plane?
  • Solid
  • Vector Form of Equation of Plane in Normal Form
  • Cartesian Form of Equation of a Plane in Normal Form
  • Distance of a Point from a Plane in Cartesian Form
  • Distance of a Point from a Plane in Vector Form
  • Points, Lines, and Planes Solved Examples
  • Points, Lines and Planes Worksheet

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Points, Lines, and Planes in Geometry

In basic geometry, fundamental concepts like points, lines, and planes form the foundation upon which more complex geometric ideas are built. Points are precise locations in space, devoid of size or dimension, represented simply by dots....

What is a Point?

A Point in geometry is defined as a location in the space that is uniquely determined by an ordered triplet (x, y, z) where x, y, & z are the distances of the point from the X-axis, Y-axis, and Z-axis respectively in the 3-Dimensions and is defined by ordered pair (x, y) in the 2-Dimensions where,  x and y are the distances of the point from the X-axis, and Y-axis, respectively. It is represented using the dot and is named using capital English alphabets. The figure added below shows a point P in the 3-D at a distance of x, y, and z from the X-axis, Y-axis, and Z-axis respectively....

Collinear and Non-Collinear Points

When 3 or more points are present on the straight line then such types of points as known as Collinear points and if these points do not present on the same line, then such types of points are known as non-collinear points....

Coplanar and Non-Coplanar Points

When the group of points is present on the same plane then such types of points are known as coplanar points and if these points do not present on the same plane, then such types of points are known as non-coplanar points....

What is a Line?

A Line in three-dimensional geometry is defined as a set of points in 3D that extends infinitely in both directions It is the smallest distance between any two points either in 2-D or 3-D space. We represent a line with L and in 3-D space, a line is given using the equation,...

Line Segment

A line segment is defined as the finite length of the line that is used to join two points in 2-D and 3-D. It is the shortest distance between two points. A line segment between two points A and B is denoted as, AB...

Mid-Point

Midpoint is defined as the point on the line segment which divides the line segment into two equal parts. Suppose we have two points A and B and the line segment joining these two points is AB and not the point P on the line is called the midpoint if it breaks the line into two equal parts such that,...

Rays

A ray is defined as a line that has a fixed end point in one direction but can be extended to infinity in the other direction. It is of infinite length. We define the ray joining points O and A and extending to infinity towards A as...

Intersecting and Parallel lines

In 2-D any two lines can either meet at some point or they never meet at some point. The lines that meet at some point are called intersecting lines. The distance between the intersecting line keeps on decreasing as we move toward the point of intersection, and at the point of intersection of these lines, the distance between them becomes zero. When two lines intersect an angle is formed between them....

Perpendicular Lines

Intersecting lines that intersect at right angles are called perpendicular lines. The angle between these perpendicular lines is always the right angle or 90 degrees....

What is a Plane?

A Plane in three-dimensional (3D) geometry is a surface such that the line segment joining any two points lies completely on it. It is the collection of all the points and can be extended infinitely in any of the two dimensions....

Solid

A solid is a 3-D concept we also called, space. We defined the solid as the extended plane that has three dimensions length breadth and height. A solid can be extended infinitely to incorporate all the space in 3-D....

Vector Form of Equation of Plane in Normal Form

The vector form of the equation of a plane in normal form is given by:...

Cartesian Form of Equation of a Plane in Normal Form

The cartesian form of the equation of a plane in normal form is given by:...

Distance of a Point from a Plane in Cartesian Form

The distance of a point P (xo, yo, zo) from a plane π: (a x + b y + c z +d = 0) in the cartesian form is defined as the length (L) of the perpendicular drawn from that point to the plane. That is calculated using the formula,...

Distance of a Point from a Plane in Vector Form

The distance of a point P having position vector from a plane π:  in vector form is defined as the length (L) of the perpendicular drawn from that point to the plane....

Points, Lines, and Planes Solved Examples

Example 1: The vector equation of the plane in 3D space which is at a distance of  5 units from the origin and normal to the vector (4 i+ 3 k) is given by?...

Points, Lines and Planes Worksheet

Identify which of the following sets of points are collinear: A(2, 3), B(5, 7), C(8, 11).Determine if the points D(4, 6), E(4, 6), and F(4, 8) lie on the same line.Given three non-collinear points, how many distinct lines can be drawn through them?Determine whether the points G(2, 3, 1), H(4, 1, 3), and I(6, 5, 2) lie on the same plane.Find the equation of the line passing through the points P(2, 1) and Q(4, 5) in slope-intercept form.Determine the distance between the points R(3, 2, 4) and S(7, 5, 8)....

Points, Lines, and Planes – FAQs

What is Geometry?...