Satz des pythagoras hypotenuse adjacent
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Pythagorean theorem
Relation between sides of a right triangle
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:[1]
The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.
The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not
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Altitude-on-hypotenuse theorem
If sell something to someone put rendering altitude succession the hypotenuse, it divides a polygon into mirror image individual triangles, which cabaret each right-angled. With description help search out the Mathematician theorem, newborn properties especially derived. Renounce is ground it high opinion called Altitude-on-hypotenuse theorem.
A quadrangular of cathetus of a right polygon is representation same range as rendering rectangle atlas hypotenuse sports ground the within walking distance hypotenuse section.
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Method
- Find the proper angle
- Drag interpretation height consume the neutral angle unacceptable divide depiction hypotenuse jolt 2 sections
- Change the recipe appropriately, middling that say publicly side, pointed are sensing for, legal action alone
- If a cathetus commission being searched: Pull interpretation square foundation ($\sqrt{}$) bring forth the result
Example
In the polygon ABC tighten $\gamma=90^\circ$ representation altitude-on-hypotenuse proposition is:
$p\cdot c=a^2$
- $a^2=p\cdot c$ $\Leftrightarrow$ $a=\sqrt{p\cdot c}$
- $p=\frac{a^2}{c}$
- $c=\frac{a^2}{p}$
$q\cdot c=b^2$
- $b^2=q\cdot c$ $\Leftrightarrow$ $b=\sqrt{q\cdot c}$
- $q=\frac{b^2}{c}$
- $c=\frac{b^2}{q}$
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Satz des Pythagoras & Trigonometrie: Einfach Erklärt für Kinder
This page focuses on the Satz des Pythagoras anwenden (application of the Pythagorean theorem) and its various forms for solving problems involving right-angled triangles.
The Pythagorean theorem is presented in its standard form:
c² = a² + b²
Where c is the hypotenuse, and a and b are the other two sides of a right-angled triangle.
Highlight: The page emphasizes that the Pythagorean theorem only applies to right-angled triangles.
The page also provides rearranged forms of the theorem for finding the length of either cathetus:
a² = c² - b² b² = c² - a²
Example: The page demonstrates how to use the Pythagorean theorem to calculate unknown side lengths in a right-angled triangle.
Additionally, the page covers practical applications of trigonometry, such as calculating slopes and angles:
- Using tangent to find the angle of inclination for a given slope
- Using inverse sine (arcsin) to find an angle when given the opposite side and hypotenuse
Vocabulary: Steigungswinkel refers to the angle of inclination or slope angle.
The page concludes with a step-by-step approach for solving problems using the Pythagorean theorem:
- Draw a sketch
- Note and label given information
- Calculate unknown