Geometry on the Web

Geometry is a study of space. Our Geometry begins with a study of shapes.

1. The simplest shape of all is a point. A Point is the centre of a dot which can be placed on a page or surface. A ‘point’ really has no size since it is the exact centre of even the smallest dot. When we draw points we draw a small dot to indicate its position.

2. A straight line can be drawn through two points.

The straight line contains an infinity of points.

The line has no beginning and no end.

It is drawn through and past the points used in its name.

It can go on forever. When we draw straight lines we use a ruler.

‘Collinear’ means ‘co-line-ar’ or ‘together on a straight line’

3. A line segment is part of a line.

A line segment has a start point and a stop point and no further.

Its length can be measured with a ruler.

4. A half-line (or ray) has a start point but no end-point.

It goes on forever in one direction past the point used to give its

name.

5. Lines, half-lines, and line segments are then linked to form shapes such as angles, triangles, squares, rectangles, parallelograms, rhombuses… and curved lines form such things as circles, parabolas, ellipses, hyperbolas, …

6. An angle is the amount of turn between two half-lines

which have the same start-point in common.

The turn of an angle is measured in either degrees or radians.

Degrees are shown on a protractor.

Radians are studied much later in school.

7. Two lines which are always the same distance apart from each other are parallel to each other.

Two lines which intersect to form a 90º angle are perpendicular to each other.

In Section 1 of observations(1) the symbols for these words are shown.

8. A triangle is a shape with 3 sides (3 line segments) forming 3 angles. There are 4 types of triangle studied – isosceles, equilateral, right angled, scalene.

Look at drawings of these triangles in Section 1 of observations(1). Can you draw a scalene triangle? (No two sides the same length! or

All three sides different lengths. Can you draw a scalene triangle which is not a right-angled triangle? How would you be sure?)

This site allows you to draw triangles.

9. A parallelogram is a shape with 4 sides. It also has opposite sides parallel.

10. A rectangle is a shape with 4 sides. It also has opposite sides parallel. A rectangle is one type of parallelogram.

11. A square is a shape with 4 sides. It also has opposite sides parallel. A square is one type of parallelogram.

When asked to draw a parallelogram most people draw one which does not look like a rectangle or a square but they could draw a rectangle or a square and be correct.

Draw a parallelogram. Draw a square parallelogram.

12. A rectangle is a parallelogram which has 4 angles of 90º.

13. A square is a parallelogram which has 4 angles of 90º. A square is one type of rectangle.

When asked to draw a rectangle most people draw one which does not look like a square but they could draw a square and be correct.

Draw a rectangle. Draw a square rectangle.

14. A square is a rectangle which has 4 sides the same length. (and opposite sides parallel, and 4 angles of 90º).

Look in Section 1 of observations(1) to see drawings of these shapes.

15. A rhombus has 4 sides the same length. So a square is also a rhombus.

A rhombus does not have to have 4 angles of 90º - a rhombus can look like a square pushed sideways.

So when asked to draw a rhombus most people do not draw a square (they draw a diamond shape) but they could draw a square and be correct.

Draw a rhombus. Draw a square rhombus.

16. A quadrilateral is the name for any drawing which has 4 sides. It could be a square or rhombus or parallelogram or rectangle or something else once it had 4 sides.

Draw a quadrilateral which is not a parallelogram of any sort.

17. A diagonal is a line segment which links opposite corners (vertices) of a shape. E.g. the diagonal of a square.

Draw a rectangle and draw one diagonal in it.

18. So far the shapes mentioned are either one-dimensional (a line), or two-dimensional (angles, triangles, … ).

19. A ‘point’ itself is zero-dimensional!

20. We live in a 3D (three-dimensional) world. Some say it is really 4-dimensional allowing for ‘time dimension’ as well, or even higher dimensions can be thought of. However, the study of Geometry at school is mostly of the 2-dimensional variety.

21. Euclid was not the first to study Geometry but he wrote 13 books on the subject and what he wrote is still the starting place for study today.

Or you can find a better summary of what Euclid did in Geometry at http://members.aol.com/bbyars1/euclid.html .

Summary:

You have seen the words:

Point, Line (straight line), Half-line, Line segment, Angle, Degrees,

Parallel, Perpendicular, Triangle, Isosceles, Equilateral, Right angled,

Parallelogram, Rectangle, Square, Rhombus, Quadrilateral, Diagonal,

Scalene, Dimension

(20 words).

Learning Stage 1:

Task 1: Learn the meaning of each of these words.

There are pictures to accompany these words and you can look at these pictures in Section 1 of observations(1) (on Words and Meanings; Angles) and Section 4 of observations(2).

The preliminary ideas about planes, points and lines are presented interactively at this teachnet site.

Look at them to reinforce the ideas which you have just met.

At this stage you can do the first Test Yourself Quiz!

Additional Investigations/Learning: (Require Java and JavaSketchpad)

(JavaSketchPad and Java if they are not installed on your computer)

Triangle for investigating the varying lengths of sides of a triangle

perpendicular for Perpendicular Lines

3 Heights for Altitudes of a Triangle

vertically opposite for Vertically Opposite Angles

Z-angles and alternate angles for Alternate Angles

Corresponding and 'Ladder' for Corresponding Angles

See cIrcumcircle and iNcircle Constructions

Bisect a line segment construction

AXIAL SYMMETRY illustration

Next page: Task 2 and Learning Stage2