Right Angles

Pythagorean Theorem: Converse Proof

PYTHAGOREAN THEOREM CONVERSE: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. Copy and paste the following link into your browser to learn more about geometric proof for the converse of the Pythagorean Theorem: https://youtu.be/7Id065L0xhU

Pythagorean Theorem: Converse Definition

The CONVERSE OF THE PYTHAGOREAN THEOREM is also true: ▶︎▶︎▶︎ For any three positive numbers a, b, and c such that a² + b² = c², there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b. An alternative statement is: ▶︎▶︎▶︎ For any triangle with sides a, b, c, if a² + b² = c², then the angle between a and b must measure 90°. This converse also appears in Euclid's own Elements (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."

Pythagorean Theorem: Definitions

The PYTHAGOREAN THEOREM, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The Pythagorean Theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "PYTHAGOREAN EQUATION": ➜ a² + b² = c², where 'c' represents the length of the hypotenuse and 'a' and 'b' represent the lengths of the triangle's other two sides.

Special Right Triangles and Pythagorean Triples

A SPECIAL RIGHT TRIANGLE is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°-45°-90°. Recognizing special right triangles can provide a shortcut when answering some geometry questions. A special right triangle is a right triangle whose sides are in a particular ratio, called the PYTHAGOREAN TRIPLES. You can also use the Pythagorean theorem, but if you can see that it is a special triangle it can save you some calculations. A PYTHAGOREAN TRIPLE consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. Copy and paste the following link into your browser to learn more about working with special right angles and Pythagorean triples: https://youtu.be/1NlekDtqT6I

Geometric Mean

When a positive value is repeated in either the means or extremes position of a proportion, that value is referred to as a GEOMETRIC MEAN (or mean proportional) between the other two values. For example. to find the geometric mean between 4 and 25, let x = the geometric mean: 4 / x = x / 25 [per definition of the geometic mean] x² = 100 [cross-products property] x = √100 x = 10 Thus, the geometric mean between 4 and 25 is 10. Copy and paste the following link into your browser to learn more about finding geometric mean using a calculator: https://youtu.be/LaezBiaqg0M

Altitude to the Hypotenuse and the AA [Angle-Angle] Similarity Postulate

AA SIMILARITY POSTULATE specifies that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar (Angle-Angle Similarity Postulate, or AA Similarity Postulate). Also, the scale factor is ratio of the lengths of two corresponding sides. The following theorem can now be easily shown using the AA Similarity Postulate. ➜ THEOREM 62: The ALTITUDE DRAWN ON THE HYPOTENUSE of a right triangle creates two similar right triangles, each similar to the original right triangle and similar to each other. Copy and paste the following link into your browser to learn more about the altitude drawn on the hypotenuse of a triangle: https://www.bing.com/videos/search?q=the+altitude+drawn+to+the+hypotenuse+of+a+triangle&&view=detail&mid=85422AE3DB66CA9E00E985422AE3DB66CA9E00E9&FORM=VRDGAR Copy and paste the following link into your browser to learn more about the AA [Angle-Angle] Similiarity Postulate: https://www.bing.com/videos/search?q=AA+Similarity+Postulate&&view=detail&mid=36A198AC14E4C046677836A198AC14E4C0466778&FORM=VRDGAR

Pythagorean Theorem: Extensions

EXTENSIONS TO THE PYTHAGOREAN THEOREM: Variations of Theorem 66 can be used to classify a triangle as right, obtuse, or acute. ➜ THEOREM 67: If a, b, and c represent the lengths of the sides of a triangle, and c is the longest length, then the triangle is... ∆ OBTUSE if c² > a² + b², and the triangle is... ∆ ACUTE if c² < a² + b². Copy and paste the following link into your browser to learn more about various extensionsto the Pythagorean Theorem: https://youtu.be/BCiH8Ike3n0

Pythagorean Theorem: Proof

The Pythagorean Theorem has been given NUMEROUS PROOFS, possibly the most for any mathematical theorem. These proofs are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean Theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound. Copy and paste the following link into your browser to learn more about geometric proof for the Pythagorean Theorem: https://youtu.be/BNCj-K2hd_k