A pretty slick trick, huh? But the man whom this math trick is named for is nearly as fascinating. Pythagoras, an ancient Greek thinker who was born on the island of Samos and lived from to B. E, was kind of a trippy character — equal parts philosopher, mathematician and mystical cult leader. In his lifetime, Pythagoras wasn't known as much for solving for the length of the hypotenuse as he was for his belief in reincarnation and adherence to an ascetic lifestyle that emphasized a strict vegetarian diet, adherence to religious rituals and plenty of self-discipline that he taught to his followers.
Pythagoras biographer Christoph Riedweg describes him as a tall, handsome and charismatic figure, whose aura was enhanced by his eccentric attire — a white robe, trousers and a golden wreath on his head.
Odd rumors swirled around him — that he could perform miracles, that he had a golden artificial leg concealed beneath his clothes and that he possessed the power to be in two places at one time. Pythagoras founded a school near what is now the port city of Crotone in southern Italy, which was named the Semicircle of Pythagoras.
Followers, who were sworn to a code of secrecy, learned to contemplate numbers in a fashion similar to the Jewish mysticism of Kaballah. In Pythagoras' philosophy, each number had a divine meaning, and their combination revealed a greater truth.
With a hyperbolic reputation like that, it's little wonder that Pythagoras was credited with devising one of the most famous theorems of all time, even though he wasn't actually the first to come up with the concept. Chinese and Babylonian mathematicians beat him to it by a millennium. This does not often happen in elementary mathematics but is quite common in more advanced topics.
Work out the details of the proof when D is to the left of C on the line AC. Which of the triangles below are right-angled triangles? Name the right angle in each case. F is the right angle.
Consider the sequence , , , , ,…. This sequence of positive real numbers is strictly increasing and is a whole number if and only if n is a perfect square such as 36 or The sequence tends to infinity, that is, there is no upper bound for. A right-angled triangle with equal side lengths is an isosceles triangle.
Next we consider the right-angled triangle with shorter sides 1 and. We can iterate this idea obtaining:. Using the above constructions it follows that a length where n is a whole number greater than 1, can be constructed using just ruler and compass see module, Constructions. This is not so, as was discovered about BC. These ideas are dealt with in more detail in the module, The Real Numbers.
Find the length, correct to 2 decimal places, of the missing side in the right triangle opposite. A cross-country runner runs 3km west, then 2km south and then 8km east. How far is she from her starting point? Give your answer in kilometres and correct to 2 decimal places. Three whole numbers that are the lengths of the sides of a right-angled triangle are called a Pythagorean Triad or Pythagorean Triple. The formula for how to generate such triples was known by about BC. With your calculator check this is a Pythagorean triple.
This was obviously not found by chance! Starting with 3, 4, 5 we can find or construct infinitely many such triples by taking integer multiples:. If a and b have a common factor then it also divides c. If we can find all primitive Pythagorean triples then we can find all triples by simply taking whole number multiples of the primitive triples. It is possible to list all primitive triples. We shall prove it using some elementary number theory including the use of the fundamental theorem of arithmetic and the use of the HCF.
The result gives a formula for all primitive Pythagorean Triads. At least one of a , b and c is odd since the triad is primitive. The square of a whole number is either a multiple of 4 or one more than a multiple of 4, hence a and b cannot both be odd. So we may assume a is odd, b is even and c is odd.
Finally if p and q are odd then a and c are even which is not the case. So the theorem is proved. Find the values of p and q corresponding to the triple , , from the clay tablet Plimpton Example 1: Solve for x.
Therefore, the length of the base x of the triangle is approximately The Converse of the Pythagorean Theorem states that, if is true, then the given triangle is a right triangle. Solve for the missing side. 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. Given the sides of a triangle equal to 16, 30, Any triangle of the form can be solved without applying long-step methods such as the Pythagorean Theorem and trigonometric functions.
The side opposite the right angle is the hypotenuse. The Pythagorean theorem is used to solve for the length of the hypotenuse. Begin typing your search term above and press enter to search.
Press ESC to cancel. Skip to content Home Why is Pythagorean theorem important? Ben Davis June 1, Why is Pythagorean theorem important? How is the Pythagorean theorem used in construction? What is interesting about the Pythagorean Theorem? How do we use Pythagoras theorem in daily life? Does 9 12 15 make a right triangle? Does 20 25 and 15 represent a right triangle? If the hypotenuse squared is shorter than the two side lengths squared and added together then the triangle is acute.
Another reason the Pythagorean Theorem is imported is it can help you find missing side lengths.
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