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Points Lines and Planes

Points Lines and Planes

Everything around us comprises geometrical shapes. The geometrically best description of how Berlin can be represented in a globe is a point!

We come across many types of lines in roads, edges of the wall, doors, and so on. The surface of the movie screen is two-dimensional and it resembles a plane.

Here, we will see the introduction for points, lines, and planes.

Identifying points, lines and planes

A point is an exact location in space. A point does not have width, length, or height and so they do not have any dimension.

Points are used to represent a particular location in diagrams and graphs. They are usually labeled in capital letters. In the diagram below, we have three points and they are labelled A, B, and C.

PointsPoints, StudySmarter Original

Lines are formed by infinite points that extend on both sides.

Unlike points, lines have length and so they are one-dimensional objects, that extend on both sides infinitely.

If there are two points A and B on the line we can represent a line by writing it as AB or BA.

Usually, when there are no points on a line it is represented by script letters, such as r, s, and t.

A line which has a start point but no endpoint is called a ray.

The ray containing two points A and B, with A as the starting point is represented by AB.

A line which has both a start and end points is called a line segment.

The segment between the points A and B is written as AB¯.

Ray, line segment and lineRay, line segment and line, StudySmarter Original

Planes can be thought of as infinitely many intersecting lines that extend forever in all directions.

Planes can have both length and width and so are two-dimensional objects. Planes are also represented by capital letters.

PlanePlane, StudySmarter Original

Types of points

In this subsection, we will learn about collinear, non-collinear, coplanar, non-coplanar points, and point of concurrency.

Given a point, we can draw an infinite number of lines that passes through that point. However, there is exactly only one line that can be drawn that passes through two given points.

In the diagram below, the only line that could be drawn through 2 points P and Q is given.

One unique line through two pointsOnly one line through points P and Q, StudySmarter Original

Suppose now we have 3 or more points, then we ask the question: does there exists a line which passes through all the given points? Depending on it we can categorise points into two types:

  1. Collinear points;

  2. Non-collinear points.

Collinear and Non-collinear points

We say 3 or more points are collinear if they all lie on a straight line.

Otherwise, they are non-collinear.

Collinear pointsCollinear points, StudySmarter Original

In the diagram above points A, B, C, and D all lie in the same line and so they are collinear points.

Non-collinear pointsNon-collinear points, StudySmarter Original

Now, in the diagram above there is no line that could be drawn connecting all the four points A, B, C, and D. Therefore, they are non-collinear points.

Now, non-collinear points open up the world of Geometry even more.

Given three non-collinear points we can draw exactly one plane which contains all of the three points. Also given a line and a point, only one plane can contain both of them. Similarly, given two parallel lines only one plane can contain all of them.

Non-collinear points form planesThree points, two lines, or a line and a point exterior to the line form a plane, StudySmarter Original

Now suppose we have 4 or more points, then we ask the question, do they exist in the same plane. Depending on it, we can categorise a set of points into

  1. Coplanar points

  2. Non-coplanar points

Coplanar and Non-coplanar points

If a set of points lie on the same plane, they are called coplanar points.

Otherwise they non-coplanar points.

Coplanar pointsCoplanar points, StudySmarter Original

If two or more lines meet at a point, it is called the point of concurrency.

Points of concurrencyPoint of concurrency, StudySmarter Original

Identify the points, collinear points, non-collinear and concurrent points from the below figure.

Example 1

Solution

The points are A, B, C, D, E, F, G and H.

The set of collinear points are {A, C, E}, {A, F, G}, and {H, F, E}. The points B and D are not collinear with another two points.

Point F is a concurrent point of the lines AG and EH.

In the diagram below we have some points in 3 dimensions.

Points in 3 dimensions

The points that lie in the same plane are A, B, C, and D. The points E and F are outside this plane.

Types of lines

As we saw before, a line extends in both directions. Lines can be straight and curved. When lines are straight, we can categorise lines as one of the three below.

  1. Horizontal line

  2. Vertical line

  3. Oblique line

Types of linesTypes of lines, StudySmarter Original

Observing the picture above, we can say that,

  • Horizontal lines go from left to right. In a cartesian diagram it runs along or parallel to the X-axis;

  • Vertical lines go up and down. In a cartesian diagram it runs along or parallel to the Y-axis;

  • Straight lines that are not vertical or horizontal are called Oblique lines.

Real-life examples

  1. Horizontal lines - The edges of the steps on the staircase.
  2. Vertical lines - A row of tall trees on highways.
  3. Oblique line - The Handel of the staircase.

When we have two lines then they either intersect or do not intersect at any point. Depending on this we have

  1. Parallel lines

  2. Intersecting lines

Parallel and intersecting lines

We say two lines are parallel if they do not have any point of intersection.

If two lines intersect, then they are intersecting lines.

When two lines intersect they intersect at a point. In particular, if the angle between the two lines is 90º, then they are called perpendicular lines.

Parallel, intersecting and perpendicular linesParallel, intersecting and perpendicular lines, StudySmarter Original

Types of planes

Similar to that of two lines we can categorise given two planes as either

  1. Parallel planes or

  2. Intersecting planes

When two planes never intersect each other they are called parallel planes.

Otherwise, they are called intersecting planes.

When two planes intersect they intersect along a line. And, similarly to lines, planes can also intersect at an angle of 90º, which are called perpendicular planes.

Parallel and intersecting planesParallel and intersecting planes, StudySmarter Original

Points, Lines, and Planes - Key takeaways

  • The point is dimensionless, line is one-dimensional and plane is two-dimensional object.
  • Numerous straight lines can be drawn with one point. Only one line can be drawn through two given points. If three or more points lie in the same line we call them collinear. Otherwise they are non-collinear points.
  • There is exactly one plane which contains 3 non-collinear points, a line and a point, and 2 parallel lines. When two or more points or two or more lines lie in the same plane they are called co-planar. Otherwise, they are non-coplanar.
  • Two lines intersect at a point. A line intersects a plane at one point. The meeting point of the two planes is a straight line.

Frequently Asked Questions about Points Lines and Planes

Points are geometrically represented by dots, and they represent exact locations in space. So, the tip of a pencil or a pen, the tip of your finger, a star at the distance, or a button may be examples of points in real life.

There are straight lines, which include horizontal, vertical and oblique lines, and curved lines.

To find the point of intersection of a line and a plane, you look for where the line meets the plane. For further precision, we would need to work with the line and plane equations.

A point is an exact place in space and is usually represented by a dot. A line extends infinitely on both sides and is generally represented by a dash. A plane extends infinitely in all directions and is usually represented by a flat surface which looks like a flat paper sheet.

A plane is geometrically represented by a flat surface like a paper sheet. We would need to work out its equation for further info on the plane.

Final Points Lines and Planes Quiz

Question

What are parallel lines?

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Answer

Two or more straight lines in the same plane which are equidistant (having the same distance between them at all points) and never intersect each other at any point are called parallel lines.

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Question

Can parallel lines contain more than two lines? 

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Answer

Yes

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Question

Perpendicular lines and parallel lines are equal.

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Answer

True

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Question

What is the line cutting the parallel lines called?

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Answer

Transversal line

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Question

Which of the following is/are the condition/s for parallel lines?

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Answer

Equal distance between lines

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Question

The gradient/slope of parallel should always be equal.

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Answer

True

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Question

For which of the following, the pair of angles are not equal?

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Answer

Corresponding angles

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Question

What are perpendicular lines?

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Answer

Two distinct straight lines which intersect at 90° are called perpendicular lines.

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Question

What is the intersecting angle measure for perpendicular lines?

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Answer

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Question

Are all the four angles of perpendicular lines measure 90°?

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Answer

Yes

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Question

The slopes of perpendicular lines are the same.

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Answer

True

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Question

What is the relation between the slopes of perpendicular lines?

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Answer

The slopes of perpendicular lines are negative reciprocal of each other.

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Question

The y-intercept in perpendicular lines are always the same.

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Answer

True

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Question

If the slopes of two lines are negative, is it possible that both lines are perpendicular?

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Answer

Yes

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Question

The line passing through (5,2) and perpendicular to line x=0 is?

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Answer

x+2y=4

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