Pythagorean Theorem Definition and Examples
The Pythagorean theorem states that 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 lengths of the other two sides. In the formula a² + b² = c², "a" and "b" are the lengths of the triangle's legs and "c" represents the length of the hypotenuse.
Pythagoras Theorem and its Proof
The Pythagorean theorem can only be applied to right-angled triangles. It is used to find the length of a missing side of a right triangle. When taking the square root of the equation, the length of the side cannot be negative.
Pythagorean Theorem Examples
For example, in a right-angled triangle where the legs are 3 and 4 units long, we can easily use the Pythagorean theorem to find the length of the hypotenuse. By applying a² + b² = c², we get 3² + 4² = 25, which equals c². When solving this equation, the hypotenuse length c is found to be 5.
Pythagorean Theorem Application in Real Life
The Pythagorean theorem finds applications in daily life, such as in architecture and construction. It can be used to solve real-world problems, such as finding the diagonals of a rectangle. In construction, the theorem can be utilized to ensure that structures are built with accurate, right-angled corners.
Basic Pythagorean Triples Worksheet and Formula
By recognizing a relationship between sides of a right-angled triangle, the Pythagorean theorem can easily determine the length of the third side. For instance, a common Pythagorean triple is 6, 8, 10 (where 6² + 8² = 10²).
Challenging Examples
In a challenging example, working out the length of the diagonal of a square with side lengths of 30 cm can be solved by applying the Pythagorean theorem. By using the formula a² + b² = c², the diagonal is found to be 50 cm. Similarly, for a pair of points (4,7) and (16, 12), the Pythagorean theorem can be used to find the length of the line connecting the two points. By applying the formula, the length is determined to be 13 units.
