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PYTHAGORAS by Mind Map: PYTHAGORAS

1. Trigonometric Ratios

2. HISTORY: The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (569 B.C.?-500 B.C.?), who was perhaps the first to offer a proof of the theorem. But people had noticed the special relationship between the sides of a right triangle long before Pythagoras. Right triangle with sides a, b, cThe Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypoteneuse, or, in mathematical terms, for the triangle shown at right, a2 + b2 = c2. Integers that satisfy the conditions a2 + b2 = c2 are called "Pythagorean triples." https://www.ualr.edu/lasmoller/pythag.htmlhttps://www.ualr.edu/lasmoller/pythag.html

3. Phocylides, Greek gnomic poet

4. Theognis of Megara, was born about 560 BC.

5. About 560 BC. He was born in Samos, Greece, and passed away in Metapontum. His actual full name is Pythagoras of Samos.

6. Books: Golden Verses, Poem of Admonition

7. Parents: Mnesarchus, Pythais

8. What is Pythagoras Theorem used for? It is used when construction workers lay foundation for the corners of a building. It is can also be used anytime where there is a right angle. Another example where the pythagoras theorem is used is when you determine the viewing size on the television screen.

9. Proof Now we start with four copies of the same triangle. Three of these have been rotated 90°, 180°, and 270°, respectively. Each has area ab/2. Let's put them together without additional rotations so that they form a square with side c. The square has a square hole with the side (a - b). Summing up its area (a - b)² and 2ab, the area of the four triangles (4·ab/2), we get c² = (a - b)² + 2ab = a² - 2ab + b² + 2ab = a² + b²

10. Proof of Pythagoras Theorem: There are actually a total of 367 proofs of the Pythagorean Theorem. 1. This is most well-known proof of the Pythagoras Theorem Proposition. This proof is the first of Euclid's 2 proofs (I.47) The underlying configuration became known under a variety of names, but it is widely known as the Bride's Chair. 2. We start with two squares with sides a and b, respectively, placed side by side. The total area of the two squares is a2+b2. The construction did not start with a triangle but now we draw two of them, both with sides a and b and hypotenuse c. As a last step,rotate the triangles 90 degrees, each around its top vertex. The right one is rotated clockwise whereas the left triangle is rotated counterclockwise. We can see that the resulting shape is a square with the side c and area c2. There are many other proofs of the Pythagorean theorem, but these 2 proofs are the more well-known ones and are simpler to understand.

11. Video explaining Pythagoras Theorem

12. Converse of the Pythagorean Theorem (click to view image): 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.

13. Using the Pythagorean Theorem to Solve Real World Problems The Pythagorean Theorem is one of the most useful formulas in mathematics because there are so many applications of it in out in the world. For example, architects and engineers use this formula extensively when building ramps: The owners of a house want to convert a stairway leading from the ground to their back porch into a ramp. The porch is 3 feet off the ground, and due to building regulations, the ramp must start 12 feet away from the base of the porch. How long will the ramp be? To solve a problem like this one, it often makes sense to draw a simple diagram showing the legs and hypotenuse of the triangle. Looking at the diagram, we can identify the legs and the hypotenuse of the triangle in the problem we need to solve. We know that the triangle is a right triangle since the ground and the raised portion of the porch are perpendicular—this means we can use the Pythagorean Theorem to solve this problem. We are given the lengths of legs a and b, so we can use that information to find the length of c, the hypotenuse.

14. Summary The Pythagorean Theorem states that in any right triangle, the sum of the squares of the legs equals the square of the hypotenuse. This Theorem is represented by the formula . If you know the lengths of any two sides of a right triangle, we can apply the Pythagorean Theorem to find the length of the third side.

15. A. Sine = Opposite/Hypotenuse B. Cosine = Adjacent/Hypotenuse C. Tangent = Opposite/Adjacent REMEMBER: TOA CAH SOH

16. Application (Jubilee and Abigail)

17. a) Using a clinometer to measure the angle of certain objects from the ground. We used the clinometer to measure the angle of the small lamp at the entrance of the sports hall from the ground. The angle we got was about 70 degrees. We also measured the angle of the tall lamp post from the ground. We got an angle of around 134 degrees. Lastly, we measured the angle of the basketball post in the sports hall from the ground, and obtained a result of 160 degrees.