Pythagoras Theorem and Proof
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Pythagoras, a Greek mathematician and philosopher born around 572 B.C., is best known for the Pythagorean theorem, which demonstrates that in a right triangle, the sum of the squares of the shorter sides equals the square of the longest side. He spent 22 years traveling to learn and established a society called the Pythagoreans, focusing on moral, political, and social life. The document explains Pythagoras's contributions to mathematics and includes proofs and applications of his theorem.
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Proof of Pythagoras Theorem
∠FBC = ∠ FBA + ∠ABC = 90° + 90° = 180°
FBC is a straight line
Since, GA // FB [Opp. sides of the square]
GA is parallel to FC
Since, BMN and AE both are perpendicular to the
same line AC
AE is parallel to BMN. As each angle of
quadrilateral AMNE is 90°, AMNE is a rectangle
Now, let ∠BAC = x
∴ ∠GAC = ∠BAE [Each 90° + x]
AG = AB [Sides of the same
square
AC = AE [Sides of the same
Square]
▲GAC = ▲BAE [By SAS]
Area of ▲GAC = area of ▲BAE
Continued](https://image.slidesharecdn.com/pythagoras-190604063720/85/Pythagoras-Theorem-and-Proof-5-320.jpg)
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Proof of Pythagoras Theorem
Area of ▲GAC = area of ▲BAE ……(i)
[Congruent triangles are equal in area]
We know, the area of a triangle is half the area of
a parallelogram if both are on the same base and
between the same parallels.
Since, GAC and square ABFG are on the same base
and between the same parallels
∴ Area of GAC=1/2 * area of square ABFG…..(ii)
Similarly, ▲BAE and rectangle AMNE are on same base
and between the same parallels
Area of ▲BAE=1/2*area of rectangle AMNE……(iii)
From equations (i),(ii) and (iii)we get:
Area of square ABFG=area of rectangle AMNE…(iv)
Similarly area of square BCHI=area of rectangle
CMND…..(v)
Therefore, sum of areas of squares ABFG and
BCHI=area of square ACDE
Therefore, AB2+BC2=AC2, hence proved.](https://image.slidesharecdn.com/pythagoras-190604063720/85/Pythagoras-Theorem-and-Proof-6-320.jpg)
Pythagoras Theorem and Proof
- 1. Pythagoras The first pure Mathematician Ronnith Nandy Presented by:
- 2. Biography Pythagoras was a Greek mathematician and a philosopher, but was best known for his Pythagorean Theorem. 01 Pythagoras He was born around 572 B.C. on the island of Samos.02 For about 22 years, Pythagoras spent time traveling though Egypt and Babylonia to educate himself.03 Pythagoras formed a brotherhood that was an exclusive society devoted to moral, political and social life. This society was known as Pythagoreans. 04
- 3. A brief history of the Pythagorean Theorem Today we write it as: a2 + b2 = c2 In the time of Pythagoras they didn't use letters yet to replace variables. And he proved that this doesn't only work for the special triangles, but for any right triangle. The sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. He studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. Instead they wrote down everything in words Pythagoras was fascinated by triangles with the special side ratios. 1 2 3 4 5 6
- 4. Verifying the Pythagorean Theorem If you count the triangles in squares a and b, the legs of the right triangle, you will see that there are 8 in each. The square on the hypotenuse of the triangle, c, contains 16 triangles. It is thought that the Babylonians saw this pattern of tiles to be a proof of the Pythagorean Theorem.
- 5. Simple PowerPoint Presentation Simple PowerPoint Presentation Simple PowerPoint Presentation Proof of Pythagoras Theorem ∠FBC = ∠ FBA + ∠ABC = 90° + 90° = 180° FBC is a straight line Since, GA // FB [Opp. sides of the square] GA is parallel to FC Since, BMN and AE both are perpendicular to the same line AC AE is parallel to BMN. As each angle of quadrilateral AMNE is 90°, AMNE is a rectangle Now, let ∠BAC = x ∴ ∠GAC = ∠BAE [Each 90° + x] AG = AB [Sides of the same square AC = AE [Sides of the same Square] ▲GAC = ▲BAE [By SAS] Area of ▲GAC = area of ▲BAE Continued
- 6. Simple PowerPoint Presentation Simple PowerPoint Presentation Simple PowerPoint Presentation Proof of Pythagoras Theorem Area of ▲GAC = area of ▲BAE ……(i) [Congruent triangles are equal in area] We know, the area of a triangle is half the area of a parallelogram if both are on the same base and between the same parallels. Since, GAC and square ABFG are on the same base and between the same parallels ∴ Area of GAC=1/2 * area of square ABFG…..(ii) Similarly, ▲BAE and rectangle AMNE are on same base and between the same parallels Area of ▲BAE=1/2*area of rectangle AMNE……(iii) From equations (i),(ii) and (iii)we get: Area of square ABFG=area of rectangle AMNE…(iv) Similarly area of square BCHI=area of rectangle CMND…..(v) Therefore, sum of areas of squares ABFG and BCHI=area of square ACDE Therefore, AB2+BC2=AC2, hence proved.
- 7. Simple PowerPoint Presentation Simple PowerPoint Presentation Simple PowerPoint Presentation Simple PowerPoint Presentation Visual Proof of Pythagoras Theorem
- 8. Total distance travelled = 21.4 + 16.3 = 37.7 miles Application of Pythagoras Theorem in Problem Solving A boat sails due East from a Harbour (H), to a marker buoy (B), 15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. Make a sketch of the journey. What is the total distance travelled by the boat? H B L 15 miles 6.4 miles 2 2 2 15 6.4LH 2 2 15 6.4LH 16.3 milesLH Example - 1
- 9. The base of the Ladder from the house is 7.3 ft Application of Pythagoras Theorem in Problem Solving A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. How far is the base of the ladder from the house? 12 ft 9.5 ft L 2 2 2 12 9.5L 2 2 12 9.5L 7.3L ft Example - 2
- 10. Simple PowerPoint Presentation Simple PowerPoint Presentation Simple PowerPoint Presentation Simple PowerPoint Presentation Water Model of Pythagoras Theorem