Pythagorova věta
matematikaCZ・4 minutes read
The Pythagorean theorem expresses that in a right-angled triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse, represented by the formula A² + B² = C². Using an example with a hypotenuse of 70 cm and one side at 46 cm, the length of the other side is calculated to be approximately 53 cm.
Insights
- The Pythagorean theorem, formulated by the ancient Greek mathematician Pythagoras, establishes a fundamental relationship in right-angled triangles, where the sum of the squares of the two shorter sides equals the square of the longest side, expressed as A² + B² = C². This principle not only serves as a cornerstone in geometry but also facilitates various practical applications in fields such as architecture and engineering.
- In a practical application of this theorem, if the hypotenuse measures 70 centimeters and one perpendicular side is 46 centimeters, the length of the other perpendicular can be calculated to be approximately 53 centimeters. This example illustrates how the theorem can be used to derive unknown lengths in real-world scenarios, emphasizing its utility beyond theoretical mathematics.
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Recent questions
What is the Pythagorean theorem?
The Pythagorean theorem is a fundamental principle in geometry that relates to right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This relationship is mathematically expressed as A² + B² = C², where A and B are the lengths of the two shorter sides, and C is the length of the hypotenuse. This theorem is widely used in various fields, including mathematics, physics, engineering, and architecture, to solve problems involving right triangles.
How do you calculate a triangle's hypotenuse?
To calculate the hypotenuse of a right triangle, you can use the Pythagorean theorem, which states that the square of the hypotenuse (C) is equal to the sum of the squares of the other two sides (A and B). The formula is expressed as C² = A² + B². To find the length of the hypotenuse, you would first square the lengths of the two shorter sides, add those values together, and then take the square root of the result. This method provides a straightforward way to determine the hypotenuse when the lengths of the other two sides are known.
What is a right-angled triangle?
A right-angled triangle, also known as a right triangle, is a type of triangle that has one angle measuring exactly 90 degrees. This right angle distinguishes it from other types of triangles, which may have acute (less than 90 degrees) or obtuse (greater than 90 degrees) angles. The side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle. The other two sides are referred to as the perpendicular sides or legs. Right-angled triangles are significant in various mathematical applications, particularly in trigonometry and geometry, and they serve as the foundation for the Pythagorean theorem.
What is the formula for finding a triangle's sides?
The formula for finding the sides of a right triangle is derived from the Pythagorean theorem, which states that A² + B² = C². In this equation, A and B represent the lengths of the two shorter sides (the legs), while C represents the length of the hypotenuse. If you know the lengths of any two sides, you can rearrange the formula to solve for the third side. For example, if you know the hypotenuse and one leg, you can find the other leg using the formula X² = C² - Z², where X is the unknown leg, C is the hypotenuse, and Z is the known leg. This formula allows for the calculation of any side of a right triangle when the other two sides are known.
How do you find the length of a triangle's side?
To find the length of a side in a right triangle, you can apply the Pythagorean theorem, which provides a method for calculating the lengths of the sides based on the relationship between them. If you have the lengths of the hypotenuse and one leg, you can use the formula X² = C² - Z², where X is the length of the unknown leg, C is the hypotenuse, and Z is the known leg. By squaring the length of the hypotenuse and subtracting the square of the known leg, you can find the square of the unknown leg. Finally, taking the square root of that result will give you the length of the unknown side. This approach is essential for solving various geometric problems involving right triangles.
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Summary
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Understanding the Pythagorean Theorem Explained
- The Pythagorean theorem, attributed to the ancient Greek mathematician Pythagoras, states that in a right-angled triangle, the sum of the areas of the squares on the two shorter sides (perpendiculars) equals the area of the square on the longest side (hypotenuse), mathematically expressed as A² + B² = C², where A and B are the lengths of the perpendicular sides and C is the length of the hypotenuse.
- In a practical example, if the hypotenuse (C) is 7 decimeters (70 centimeters) and one perpendicular (Z) is 46 centimeters, the length of the other perpendicular (X) can be calculated using the formula X² = C² - Z². Converting 7 dm to centimeters gives 70 cm, leading to the equation X² = 70² - 46², which simplifies to X² = 2784.
- To find the length of the perpendicular X, the square root of 2784 is calculated. Rounding 2784 to 2800 for easier calculation, the square root of 28 is approximately 5.2. Adjusting for the two zeros gives a final result of 52.9 cm, which can be rounded to 53 cm, confirming that the length of the second perpendicular is 53 cm.
