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An Important Theorem In Mathematics

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Published in: Mathematics
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An Important Theorem in Mathematics.

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  1. The Learning Hall Mathematics Class IX-X sep 2, 2018 1. Prove the PYTHAGORAS THEOREM ' In a Right Angled Triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides' A. GIVEN c D Let ABC be a Right Angled Triangle, right angled at B R.T.P CONST. PROOF. AC2 = AB2 + BC2 Let BD be drawn perpendicular to AC From the above diagram we can see that As ABC, ABD and BCD are right angled triangles Considering As ABC and ABD Z ABc= Z BAC = = 900 (By Construction) Z ADB Z BAD (Common)
  2. Therefore, Remaining BCA = By similarity of triangles, DC).AC = AB2+ Remaining A ADBN A ABC ( By AA Similarity Criterion) Similarly, A BDCN A ABC ( By AA Similarity Criterion) From Eq. (i) Z ABD (ii) AD/AB = AB/AC (Since corresponding sides are proportional, in similar triangles) Hence, AD.AC = AB2 From Eq. (ii) (iii) DC/BC = BC/AC (Since corresponding sides are proportional, in similar triangles) Hence, DC.AC = BC2 Adding (iii) & (iv), we get AD.AC+ DC.AC= AB2+BC2 That is, or, AC.AC = AB2+ BC2 or, AC2 = AB2+BC2 (iv) Therefore, AB2 + = AC2
  3. 2. Prove the CONVERSE OF 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'. QR = BC Considering Eq. (ii) & (iii), we get B. GIVEN c Let ABC be a triangle in which AC2 = AB2+ R.T.P CONST. PROOF. Z ABC = 900 Let us construct A PQR right angled at Q, such that (ii) In A PQR, we have PR2 = PQ2 + QR2 PR2 = AB2 + BC2 (By Pythagoras Theorem) (iii) (iv)
  4. or, From Eq. (i) & (iii), we get AB = PR BC = QR AC2 = PR2 AC = PR (Taking square root on both sides) (v) Now, in As ABC and PQR (ii) (v) Hence A ABC = Hence, Z B A PQR (By SSS Congruency) (By CPCT) But Z Q = 900 (By Construction) Hence, 3. A. Z B -900 What are Pythagorean Triples ? Give some examples A Pythagorean Triple consists of three positive integers a, b & c such that a2 + b2 = c2 Such a triple is commonly written as {a,b,c} Some well known examples are c. { 5, 12, 13 } { 7, 24, 25 } { 8, 15, 17 } { 9, 40, 41 }

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