Pythagoras Theorem

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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

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