Contents:

1. Scale triangles                                          page 3

2. Isosceles triangle                                    page 24

3. Equilateral triangle                               page 28

4. Rectangular triangle                            page  32                         

       Scale triangles

Ana Francisco

Agrupamento de Escolas de Pombal

Portugal 

Tales Theorem

Tales Theorem

Tales' theorem is a geometry theorem that states that, in a plane, the intersection of parallel lines, by transverse lines, form proportional segments.

Pyramid Challenge Quéops

There are many stories about Tales deeds. For example, challenged by a pharaoh's scribes during his passage through Ancient Egypt, Tales determined the height of the Pyramid Cheops, with the help of sunlight, as illustrated in the figure.

Tales' theorem is an important tool for geometry, as it assists in calculating inaccessible distances and in triangle similarity relationships.

The Tales Theorem has many applications in everyday life, constituting an important tool of Geometry in the calculation of inaccessible distances and in relationships involving similarity between triangles. The best way to visualize the applicability of the Theorem proposed by Tales of Miletus is through a few examples.

Example 1

Calculate the length of the bridge that should be built over the river, according to the following scheme.

According to the figure we have an ABC triangle and the SEGMENT DE dividing the triangle, being formed the triangle ADE. The information we have is the measurements of the following segments: AD = 10m, AE = 9m, EC = 18m and DB = x. The DB value will be determined through the Tales Theorem which says: "parallel lines cut by transverse form proportional segments." In this way, we can establish the following relationship:

The bridge will be 20 meters long.

Example 2

                            This part of the Map can be represented as follows:

Problem 3

One sunny day, Ioana, whose height is 1.6 m, decided to find out how tall the tree in the yard of her house is. She left the tree and walked 12 m along it, until the tip of her shadow overlapped just above the tip of the tree's shadow. Determine the height of the tree, knowing that Ioana's shadow was 6 m long.

Mihaela

Gaudeamus High School

Republic of Moldova

Problem 4

   A pine has grown between two buildings A and B. Determine the height of the pine, knowing that its peak V, belongs to the lines MR and NQ and MQ = 60m, and NR = 40m.    

Mihaela

Gaudeamus High School

Republic of Moldova

Problem 5

 Isosceles triangle

Problem 1

In an isosceles triangle, the sum of the measures of two angles is 110 °. Determine the measures of the angles of the triangle.

proposed by Arina, solved by Zlata

Gaudeamus High School, Republic of Moldova

Problem 1

proposed by Mihaela

                 Andreea

  Gaudeamus High School

     Republic of Moldova

    Equilateral triangle

Problem 1

Three neighbors decided to dig a well so that it was located at the same distance from each house. Determine how far from each house the fountain is, if the houses are the vertices of an equilateral triangle with a side of 6 km (√3≈1.7)

Andreea

Gaudeamus High School

Republic of Moldova

   Rectangular triangle

 Pythagoras' Theorem

In a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

Example 1.

Determine the length d of the loading ramp in the image.

Gaudeamus High School

Moldova

Example 2.

The height of the house is 5 m and the distance from the house to the foot of the stairs is 2 m. What length should the ladder reach to the nest under the eaves?

Bogdana

Gaudeamus High School

Moldova

Example 3

A gas distribution system must be installed in such a way that it is located at the same distance from three blocks. Determine this distance if the blocks are the vertices of a right triangle with a catettel of 15 km and 

8 km.

                        Mihaela

        Gaudeamus High School

            Republic of Moldova

Example 4

The right triangle shown below has an area of 25. Find its hypotenuse.

Zlata

Gaudeamus High School

Republic of Moldova

          30 degree angle theorem

In a right triangle with an angle of 30 degrees, the length of the leg opposite the angle of 30 degrees is equal to half the length of the hypotenuse.

      Gaudeamus High School, Moldova

An observer sees an airplane flying at an altitude of 6000 m at an angle of 30 degrees. 

How far is the plane from the observatory?

                                Example 1

                                           Mihaela

                        Gaudeamus High School, Moldova

Example 2