Golden Rules and Geometrical places

This week we will start with some constructions. Our materials needed for drawing are:

  • Pencil or mechanical pencil (US)/propelling pencil (UK) with 4H and HB leads
  • Rubber
  • Straight ruler
  • Set squares: 45º set square and 60º set square
  • Compass or pair of compasses with a nail fair for sharpening it

Do you know what can you do with these instruments?. You can define points and draw straight lines and circumferences. These operations are based on the 5 golden rules:

  1. To draw a line you need two points
  2. The intersection between two lines is a valid point for future operations.
  3. The intersection between a line and a circumference arc is a valid point for future operations
  4. The intersection between two circumference arcs is a valid point for future operations
  5. To draw a circumference you need a centre and a radio.

We have to follow these rules to solve our geometrical exercises. If we think of any solution that doesn’t obey them, it will be incorrect in this “Geometry Game.”

The best example of a solution that does not follow the golden rules is when you try to draw a tangent line using one point on the circumference. Can’t you simply bring the ruler to the circumference and draw the tangent line? No, because it would be made by guessing because you are using only one point to draw the line. According to the first golden rule, you need two points to draw a line.

Today we will see what a geometrical place is: it is a set of points that follow the same property or attribute. This property could be:

* A set of shops with the same price for a product

* A set of pupils with the same colour of hair

You can imagine many examples, but in Geometry we are going to study these geometrical places:

  • Perpendicular bisector: a geometrical place that is the same distance to each of the points (that is equidistant from the points). In other words, the distance from the perpendicular bisector to point A is the same as the distance from the perpendicular bisector to point B. I would like to point out that you don’t need to draw the segment although I know that in many books it is explained as: “A line which cuts a line segment into two equal parts at 90°” or “It is a line segment perpendicular to the segment AB and passing through the midpoint M of it”
  • Angle bisector: a geometrical place where the points are the same distance between the two straight lines of an angle. It is the line that cuts the angle into two equal halves so that the distance from the line to one side of the angle is the same as the distance from the line to the other side of the angle.
  • Circumference: a geometrical place where the points are all the same distance from another point, called the centre
  • Capable arc: a geometrical place where the all points see a segment from the same angle.

There are also other geometrical places such as the conic section (ellipses, parabola and hyperbola), among others.

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