# UNIT 02: FUNDAMENTAL CONSTRUCTIONS

## Thales’ theorem:

If two lines which intersect each other and are in the same plane are cut by parallel lines, segments determined in one line are proportional to the segments determined in the other line.

We use the Thales theorem to divide segments into equal or proportional parts.

## Right triangle altitude theorem:

In Geometry, we say that two or more figures are **“similar”** when all of their **angles** are **equal** so their **sides** are **proportional**.

The **right triangle altitude theorem** or **geometric mean theorem** is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that in a right triangle, the altitude on the hypotenuse is the geometric mean to the two segments into which the hypotenuse is divided.

## Golden ratio:

Here you can see a video created by KhanAcademy to learn more about **Golden Ratio**

In mathematics and the arts, two quantities are in the **golden ratio** if their ratio is the same as the ratio of their sum to their maximum. The golden ratio is also called the golden section (Latin: **sectio aurea**) or golden mean. The **golden number** is an irrational number and it is represented by the Greek letter in honor to Greek sculptor Phidias.

Expressed algebraically, for quantities *a* and *b* with *a* > *b*

Where the Greek letter phi () represents the golden ratio. Its value is:

In this unit you will learn how to get this ratio graphically:

## To learn more….

*There is considerable agreement that Thales was born in Miletus in Greek Ionia in the mid 620s BCE and died in about 546 BCE, but even those dates are indefinite. Greek philosopher who is considered the founder of Greek science, mathematics, and philosophy. He visited Egypt and probably Babylon, bringing back knowledge of astronomy and geometry. He invented deductive mathematics. To him is attributed Thales’ theorem. It is also attributed to Thales the prediction of a Solar Eclipse and more theorems.*

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