Session 1: What is Geometry? – Points, Lines, Planes and Solids

The word ‘geometry’ is derived from the Greek words ‘geo’ meaning ‘earth’ and ‘metria’, meaning ‘measuring’. Solid objects, such as yourself, are three-dimensional. Thus, the world we live in is three-dimensional. To better understand why your world is three dimensional, consider zero, one and two dimensions. 

Point

A point has a dimension of zero. In mathematics, a point is the location and is represented as a dot with no size (no length or width). Points are usually indicated with a capital letter, like point A. Other basic elements of geometry are based on one or more points. 

Line, Line Segment and Ray

A line has a dimension of one. It has a length.

A number line is an example of a line, which is like an infinitely long line segment without any endpoints. You only need a single number to describe a point on a number line (in the interactive below, the location of point A is -2). Remember that, by definition, a line is straight.

A line segment is a portion of a line with two endpoints. A ray is a portion of a line with one endpoint. Line segments are named by their endpoints, and rays are named by their endpoint and another point. In each case, a segment or ray symbol is written above the points. Below, the line segment is , and the ray is .

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Plane

A plane has a dimension of two. It can be compared to a line segment because it has length and width but no height. 

A plane surface extends indefinitely in all directions. The surface of a wall and the top of a table are examples of a plane. 

Since a plane extends indefinitely in all directions, we cannot draw a plane on paper. In practice, only a portion of a plane is drawn, and usually, it is represented by a rectangle or a parallelogram. A plane is named using three or more points that are not on the same line.

Solid

A solid has a dimension of three. It has length, width and height.

 

Intersecting and parallel lines

When two lines intersect, at least one angle will be formed. The common point is called the point of intersection. 

When two lines are parallel, no angle is formed, and they do not have any common point, i.e., they do not intersect. 

Perpendicular Lines

Perpendicular lines are lines that intersect at right angles. They are denoted by the symbol ⊥, e.g.  or .   

Collinear Points

Given three randomly located points, it is unlikely that the same line will pass through all three points. If three points lie on the same line, they are called collinear .

Thus, three or more points are collinear if the line passing through any two points passes through other points.

Concurrent Lines

Similarly, given three random lines, it is unlikely that all three lines will intersect at the same point.

When three or more lines do intersect at a point, the point is called a point of concurrency.

Incredibly, the three angle bisectors, medians, perpendicular bisectors and altitudes are concurrent in every triangle.

What is Geometry? – Examples

Example 1

i. Name three points from the three-dimensional figure below. How many dimensions does each point have?

Any points that make up this prism will work. These points are called vertices. For example, point A, point B, point C. Although the rectangular prism is three-dimensional, each point is simply a location in space and has zero dimensions. 

 ii. Name three line segments from the figure above. How many dimensions does each line segment have?

Any line segments that make up this prism will work. These line segments are called edges—for example, . Although the figure is three-dimensional, each edge is only a line segment, and so is one-dimensional. 

iii. Name three sides from the figure above. How many dimensions does each side have?

Any sides that make up this prism will work. As part of a three-dimensional figure, they are two-dimensional shapes called faces. For example, ABCD (the top face), BCGF (the right side), CGHD (the front face).

Example 2

From the figure given above, name the following:

i. Pair of parallel lines

ii. Lines intersecting at B

iii. Lines intersecting at C

iv. Concurrent lines and their point of concurrence

i. Lines o and p are parallel lines.

ii. Lines intersecting at point B are m, p and q.

iii. Lines intersecting at point C are l, n and q.

iv. Lines m, p and q are concurrent with B as the point of concurrence. 

Lines l, n and q are concurrent with C as the point of concurrence. 

 Remember this!

• A point is the location and is represented as a dot with no size (no length or width).
• A line has a dimension of one. It has a length (no width or thickness).
• A line segment is a portion of a line with two endpoints. 
• A ray is a portion of a line with one endpoint. 
• A plane has a dimension of two. It has length and width (no thickness).
• A solid has a dimension of three. It has length, width and height. 
• Intersecting lines are lines that meet or cross at a common point. 
• Parallel lines are lines in the same plane that never intersect. 
• Perpendicular lines are lines that intersect at right angles. 
• Three or more points are collinear if they lie on the same line. 
• Three or more lines in a plane are concurrent if they pass through the same point, which is called their point of concurrence.