1. PYTHAGORAS AND THALES THEOREMS
1. SIMILAR FIGURES
Two figures are SIMILAR if they have the same shape but different size. Two triangles are similar when
they have equal angles and proportional sides.
2. THALES THEOREM
A theorem is a discovery we get by reasoning. There are two very important theorems in Geometry:
Thales theorem and Pythagorean theorem. Thales of Miletus was a wise man from Ancient Greece (VI
Century BC).
The property described by Thales about Geometry states the following:
Parallels through several straight lines form proportional segments. That means the triangles we get
are similar: they have equal angles and proportional sides.
If two triangles have a common angle and parallel opposite sides to the angle then they are similar. In
this case we say that they are in Thales position.
To check that two right triangles are similar, it is enough if one of their acute angles is equal or two of
their corresponding sides are proportional:
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2. This property is very useful to solve many problems in real situations:
PROBLEMS:
1. Leticia's cat has climbed on top of a post. Leticia can see its cat reflected in a puddle. If her eyes'
height is 1.44 m, how tall is the post?
Solutions: 3.6 m
2.- Calculate the height of the tallest tree:
Solutions: 45.8 m
4. PYTHAGORAS THEOREM
A right triangle has a right angle (90º) and two acute angles. The sides of a right triangle are called:
hypotenuse (a) and catheti or legs (b and c). The hypotenuse is the side opposite the right angle and is
the largest side of the triangle. The legs or catheti (singular: cathetus) are the sides opposite the acute
angles and are the shorter sides of the triangle.
The theorem establishes that:
In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.
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3. Three numbers that verify this equality are called Pythagorean triple.
Examples
1. Determine whether the following triangle is a right triangle:
The legs of a right triangle measure at 3 m and 4 m. What is the length of the hypotenuse?
The hypotenuse of a right triangle is 5 m and one of its legs is 3 m. What is the length of the other leg?
Problems
1. A ladder 10 m long is leaning against a wall. The foot of the ladder is 6 m from the base of the
wall. At what height does the ladder rest on the wall?
2. Find the diagonal of the rectangle:
3. Find the perimeter and area of the right trapezoid:
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4. P = 8 + 6 + 12 + 6.32 = 32.32 cm
4. The perimeter of an isosceles trapezoid is 110 m, the bases are 40 and 30 m respectively.
Calculate the length of the non-parallel sides and the area.
5. Find the area of the regular pentagon:
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