The Pythagorean Theorem is a fundamental principle in geometry, discovered by the ancient Greek mathematician Pythagoras. It relates to right-angled triangles, stating a specific relationship between the lengths of the sides of such triangles. The theorem is widely used in various mathematical and practical applications.
The Pythagorean Theorem Formula
In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This relationship is 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.
How to Use the Pythagorean Theorem
- Identify the lengths of two sides of a right-angled triangle.
- Apply the theorem: square these lengths and add them together.
- To find the length of the third side, take the square root of the sum if finding the hypotenuse, or use algebraic manipulation if finding a shorter side.
Examples of Using the Pythagorean Theorem
Example 1: In a right-angled triangle with sides 3 cm and 4 cm, find the length of the hypotenuse. Applying the theorem: 3² + 4² = 9 + 16 = 25. The hypotenuse is √25 = 5 cm.
Example 2: If the hypotenuse of a right triangle is 13 cm and one side is 5 cm, find the other side. Using the theorem: 13² – 5² = 169 – 25 = 144. The other side is √144 = 12 cm.
Importance of the Pythagorean Theorem
The Pythagorean Theorem is crucial in geometry for determining distances and measurements. It has practical applications in construction, navigation, architecture, and various fields of science and engineering.
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