Group of Pythagorean theorems. To establish relationships in the right triangle, it is necessary to know a group of theorems of great importance. The deduction of these relationships is achieved from the Similarity of Triangles .
Summary
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- 1 Applications
- 2 Theorem of heights
- 1 Demonstration
- 3 Catholes Theorem
- 4 Pythagorean Theorem
- 5 Reciprocal of the Pythagorean theorem
- 6 Source
Applications
There is a group of relationships that are true only in right triangles, which have somehow to do with the height relative to the hypotenuse .
These will help us to solve Triangle Similarity exercises.
Heights theorem
In any right triangle, the square of the length of the height relative to the hypotenuse is equal to the product of the lengths of the segments that the height determines on the hypotenuse.
Demonstration
If in the right triangle ABC, where angle C = 90º, the height relative to the hypotenuse is plotted, two triangles similar to ∆ ABC are obtained, which are the ∆ ACD and ∆ BCD. ( p and q are the segments that determine the height above the hypotenuse).
Effectively:
Therefore: Δ ABC ~ Δ ACD ~ Δ BCD by transitive character.
Homologous sides in these triangles :
Between the homologous sides of these three similar triangles, we can establish some proportions and from them, deduce these theorems that take place in any right triangles.
Catholes Theorem
According to the height theorem it is true that: h² = p · q . This equality is obtained in the following way:
The Δ ACD ~ Δ BCD holds that:
Of the two segments that the height determines on the hypotenuse, the one corresponding to a specific leg is the one that has a common end with it.
In any right triangle, the square of the length of each leg is equal to the product of the length of the hypotenuse times the length of the hypotenuse segment] corresponding to the leg.
Pythagoras theorem
Also, according to the height theorem, it is true that: b² = c · q and a² = c · p
Indeed, from Δ ACD ~ Δ ABC we have to:
From Δ BCD ~ Δ ABC we have to:
In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
Reciprocal of the Pythagorean theorem
The Pythagorean theorem is widely used, if we apply the legs theorem with it, we will deduce its reciprocal in the following way:
Catholes Theorem:
- b² = c · q
- a² = c · p
Adding member to member these equalities, we obtain:
a² + b² = c · p + c · q
a² + b² = c (p + q) (1) applying the distributive property.
But c = p + q (2) by sum of segments.
And substituting (2) in (1) we have:
If for the sides a , b and c of a triangle it is true that a² + b² = c² , then the triangle is right and its hypotenuse is
