Pythagoras' Theorem Group A

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Pythagoras' Theorem Group A by Mind Map: Pythagoras' Theorem Group A

1. Pic 1

2. History of Pythagoras

2.1. influenced by Pherekydes (philosophy), Thales (mathematics and astronomy) and Anaximander (philosophy, geometry)

2.2. Left Samos (birthplace) for Egypt in about 535 BC to study with the priests in the temples there

2.2.1. Many of the practices of the society he created later in Italy can be traced to the beliefs of Egyptian priests

2.2.2. When Persia invaded Egypt, Pythagoras was taken prisoner and sent to Babylon, where he met the Magoi (priests) who taught him sacred rites

2.2.2.1. He returned to Samos and started a school called the Semicircle. However, his teaching methods did not appeal to the authorities of Samos and hence Pythoragas decided to move.

2.2.2.1.1. He finally settled in Crotona (in Southern Italy) around 518 BC, and founded a philosophical and religious school where his followers lived and worked

3. Proof of Pythagoras' Theorem (refer to link for images)

3.1. Pythagoras' proof

3.1.1. The area of the first square is (a+b)^2 or 4(1/2ab)+ a^2 + b^2 and the area of the second square is given by (a+b)^2 or 4(1/2ab) + c^2.

3.1.1.1. Therefore we can obtain 4(1/2ab) + a^2 + b^2 = 4(1/2ab)+ c^2. We can subtract 4(1/2ab) from both sides to get a^2 + b^2 = c^2, which is the Pythagorean theorem

3.2. Bhaskara's 1st proof

3.2.1. In the square diagram, area of the square=c^2.

3.2.1.1. When we calculate each of the shapes in the square separately, Area of the blue triangles = 4(1/2)ab Area of the yellow square = (b-a)^2 Area of the big square = 4(1/2)ab + (b-a)^2 = 2ab + b^2 - 2ab + a^2 = b^2 + a^2

3.2.1.1.1. Thus it shows that the area of the square = c^2 = a^2 + b^2, proving that the Pythagorean theorem is correct

3.3. Bhaskara's 2nd proof

3.3.1. Begin with a right triangle and then he drew an altitude on the hypotenuse. From here, use he properties of similarity.( refer to picture in the website) Prove that triangles ABC and CBE are similar. It follows from the AA postulate that triangle ABC is similar to triangle CBE, since angle B is congruent to angle B and angle C is congruent to angle E. Thus, since internal ratios are equal s/a=a/c. Multiplying both sides by ac we get sc=a^2.

3.3.1.1. Now show that triangles ABC and ACE are similar. As before, it follows from the AA postulate that these two triangles are similar. Angle A is congruent to angle A and angle C is congruent to angle E. Thus, r/b=b/c. Multiplying both sides by bc we get rc=b^2.

3.3.1.1.1. Now when we add the two results we get sc + rc = a^2 + b^2. c(s+r) = a^2 + b^2 c^2 = a^2 + b^2, concluding the proof of the Pythagorean Theorem.

3.4. Garfield's proof

3.4.1. Find the area of the trapezoid by using the area formula of the trapezoid. A=(1/2)h(b1+b2) area of a trapezoid In the above diagram, h=a+b, b1=a, and b2=b. Area =(1/2)(a+b)(a+b) =(1/2)(a^2+2ab+b^2).

3.4.1.1. Fnd the area of the trapezoid by summing the area of the three right triangles. The area of the yellow triangle is A=1/2(ba). The area of the red triangle is A=1/2(c^2). The area of the blue triangle is A= 1/2(ab). The sum of the area of the triangles is 1/2(ba) + 1/2(c^2) + 1/2(ab) = 1/2(ba + c^2 + ab) = 1/2(2ab + c^2).

3.4.1.1.1. Since, this area is equal to the area of the trapezoid we have the following relation: (1/2)(a^2 + 2ab + b^2) = (1/2)(2ab + c^2). Multiplying both sides by 2 and subtracting 2ab from both sides we get a^2 +b^2 = c^2

4. What is the Pythagoras Theorem?

4.1. In mathematics, the Pythagorean theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the side opposite the right angle is equal to the sum of the squares of the other two sides.

4.2. The theorem states that "The square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs"

5. Applications of Pythagoras' Theorem

5.1. Architecture and Construction

5.1.1. Navigation

5.1.1.1. Surveying