Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Pythagoras Theorem

13,552 views

Published on

Published in: Education, Technology, Spiritual
  • Be the first to comment

Pythagoras Theorem

  1. 1. Pythagorean Theorem T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
  2. 2. Pythagoras theorem  The Pythagorean theorem is related to the study of sides of a right angled triangle.  It is also called as Pythagoras theorem.  The Pythagorean theorem states that,  In a right triangle (length of the hypotenuse)2 = {(1st side)2 + (2nd side)2} a b cc2 = a2 + b2
  3. 3.  In a right angled triangle three sides: Hypotenuse, Perpendicular and Base. The base and the perpendicular make an angle is 900.So, according to Pythagorean theorem: Pythagoras theorem (Hypotenuse)2 = (Perpendicular)2 + (Base)2 Pythagoras Theorem Proof: p b h A B C Given: Δ ABC is a right angled triangle where <B = 900 And AB = P, BC= b and AC = h To Prove: h2 = p2 + b2
  4. 4. Construction : Put a perpendicular BD from B to AC , where AD = x and CB = h-x , p b h A B C D x (h-x) Proof : Consider the two triangles Δ ABC and Δ ABD, these two triangles are similar to each other because of AA similarity. This is because both the triangle have a right angle and one common angle at A. In Δ ABC and Δ BDC both are similar So by these similarity, = = p h x p b h (h-x) b AND Pythagoras theorem
  5. 5. = = p h x p b h (h-x) b AND P2 = h * x and b2 = h (h – x) Adding both L.H.S. and R.H. S. P2 + b2 = h.x + h (h - x) Or P2 + b2 = h.x + h2 – h.x h2 = p2 + b2 Pythagoras theorem p b h A B C D x (h-x)
  6. 6. converse of the Pythagoras Theorem  In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. A B C Given: In a triangle ABC in which AC2 = AB2 + BC2 To prove: ∠ B = 90
  7. 7. Construction: A Δ PQR right angled at Q such that PQ = AB and QR = BC (figure) Proof: from Δ PQR, we have: PR2 = PQ2 + QR2 (Pythagoras Theorem, as ∠ Q = 90 ) or, PR2 = AB2 + BC2 (By construction) ……………. (i) converse of the Pythagoras Theorem P Q R A B C
  8. 8. But given, AC2 = AB2 + BC2……………………. (ii) From (i) and (ii) AC = PR …………………………….(iii) Now, in Δ ABC and Δ PQR, AB = PQ …………………(By construction) BC = QR ………………...(By construction) AC = PR …………….Proved in (iii) above P Q R A B C converse of the Pythagoras Theorem
  9. 9. So, Δ ABC ≅ Δ PQR …………….(SSS congruence) Therefore, ∠ B = ∠ Q ...........................(CPCT) But, ∠ Q = 90 …………………..(By construction) So, ∠ B = 90 Proved P Q R A B C converse of the Pythagoras Theorem
  10. 10. The End Call us for more information: www.iTutor.com 1-855-694-8886 Visit

×