# Tangencies and linkages

In Tangencias by José Antonio Cuadrado you can find help for next exam and next exercises. It is a fantastic resource where you can review and continue learning.

# Golden section with Donald

If there is anything you should try to avoid whenever possible, it is learning configurations by memory without understanding them, so that if your memory fails, you do not know how to solve the exercise. This is true in the case of the pentagon, which we might not be excited about to begin with, even though a great beauty is hidden in its properties. If we carefully examine the figure rather than simply learning without thinking, we will see that it is more interesting than we originally thought.

In Bachillerato, we always analyze in detail the construction of the pentagon and we check how THE SIDE IS THE GOLDEN SECTION OF THE DIAGONAL. I know that in 3rd ESO, for some, this still sounds strange, but you should pay attention to it because we will work on it in class. Now, in order to know what this “golden section” is, I’m going to hand it over to a good friend…

This is a part of the film Donald in Mathmagic World, where Donald join Pythagoras’ secret society, which distinguished its initiated with the symbols of the pentagram. Sum up of the film.

Click here to watch the whole film.

Do you want to know more about golden section and Fibonacci numbers? Visit Nature by numbers by Cristóbal Vila.

# Poligons and symmetry

After a little review, we are going to begin with some 3rd ESO content: regular polygons with a given side: triangle, square, pentagon, hexagon, heptagon, octagon and nonagon.

Although I will give you some notes, you can see two interesting resources (in Spanish): Educación Plastica.net: web by Fernando Ruiz de Lejarazu. At “trazados geométricos” you can see information and games about polygons.

Remember: polygons obey different symmetries depending on the number of their sides:

• If the number of sides is EVEN, the polygon has a radial symmetry. Its radial centre is the centre of the circumference around it.
• If the number of sides is ODD, the polygon has an axial symmetry. The vertices are opposite of the middle point of the opposite side.

Surf and practise on Ritmo y Simetría en la composición plástica by Mª Luisa Bermejo, winner of the contest Materiales Curriculares 2005. You can explore and have fun with this resource!

# 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.

# Geometry

This year we are beginning with Geometry in 3rd ESO.

I hope that you enjoy this subject and find it very interesting.

Geometry (geo=earth, metry=measurement, from Latin) was born in Egypt many centuries ago. It was born as a necessity beacuse the Nile river grew every year, flooding the land and destroying property boundaries.

These boundaries had to be re-established for the ROPE-STRETCHERS

who worked with a rope with twelve knots in order to build up right triangles of 3, 4 and 5 spaces in each side. We know about Geometry thanks to the Greeks, who conquered many territories, including Egypt, where they learned about Geometry. Greek people, especially a man named Euclid, wrote many geometric treatises. Due to this, plane geometry (two dimensions) is called Euclidean geometry.

In 1st ESO you studied some parts of plane geometry. Here are some basic things you might need to review:

Pay attention to the pronuntiation: