Pythagoras Theorem

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  First Course in Euclidean Geometry, A

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter- 3 Pythagorean Theorem and Thales' Theorem Pythagorean theorem The Pythagorean theorem: The sum of the areas of the two squares on the legs ( a and b ) equals the area of the square on the hypotenuse ( c ). In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle ( right-angled triangle ). In terms of areas, it states: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). The theorem can be written as an equation relating the lengths of the sides a , b and c , often called the Pythagorean equation : where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides. ________________________ WORLD TECHNOLOGIES ________________________ These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths . Tobias Dantzig refers to these as areal and met-ric interpretations. Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally by Descartes in his work La Géométrie, and extending today into other branches of mathematics. The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described below, including extension to many-dimensional Euclidean spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n -dimensional solids.

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  Euclidean and Projective Geometry

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 3 Pythagorean Theorem and Thales' Theorem Pythagorean theorem The Pythagorean theorem: The sum of the areas of the two squares on the legs ( a and b ) equals the area of the square on the hypotenuse ( c ). In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle ( right-angled triangle ). In terms of areas, it states: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). The theorem can be written as an equation relating the lengths of the sides a , b and c , often called the Pythagorean equation : where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides. ________________________ WORLD TECHNOLOGIES ________________________ These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths . Tobias Dantzig refers to these as areal and metric interpretations. Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally by Descartes in his work La Géométrie, and extending today into other branches of mathematics. The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described below, including extension to many-dimensional Euclidean spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n -dimensional solids.

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  Theoretical Introduction to Mathematical Theorems, A

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  Pythagoras, on the other hand, was one of the first to grasp numbers as abstract entities that exist in their own right. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power. Popular references to Pytha-goras' theorem in literature, plays, musicals, songs, stamps and cartoons abound. Other forms As pointed out in the introduction, if c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, Pythagoras' theorem can be expressed as the Pythagorean equation: or, solved for c : If c is known, and the length of one of the legs must be found, the following equations can be used: or The Pythagorean equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle, the law of cosines reduces to the Pythagorean equation. ________________________ WORLD TECHNOLOGIES ________________________ Proofs This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs. Proof using similar triangles Proof using similar triangles This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles. Let ABC represent a right triangle, with the right angle located at C , as shown on the figure. We draw the altitude from point C , and call H its intersection with the side AB .

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  How to Think Like a Mathematician

  A Companion to Undergraduate Mathematics

  • Kevin Houston(Author)
  • 2009(Publication Date)
  • Cambridge University Press

    (Publisher)

  So in this chapter we will pull apart the theorem and its proof, we’ll see a converse for it and also a generalization. Statement of Pythagoras’ Theorem As you are a budding mathematician, you probably have a better idea than a non- mathematician of what the statement is, but here it is again. Theorem 19.1 For a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other sides. Exercise 19.2 Use the ideas from Chapter 16, How to read a theorem, to analyse the theorem. Compare your analysis with the one given below. Study of the theorem We now analyse the theorem as though we were meeting it for the first time. Obviously we would check what all the words mean, for example, what is a hypotenuse? This is fairly obvious, but what about the other techniques in Chapter 16? We shall apply them now. 126 Study of the theorem 127 Draw in this line Figure 19.1 Any triangle gives two right-angled triangles Find the assumptions and conclusions The statement given for Pythagoras’ Theorem is a good example of a statement not in the form ‘A =⇒ B ’ or ‘if . . ., then . . .’. We can rewrite it in this form in a number of ways. For example, ‘If T is a right-angled triangle with sides a, b and hypotenuse c, then c 2 = a 2 + b 2 .’ This make it obvious that the assumptions concern all right-angled triangles: ‘T is a right-angled triangle with sides a, b and hypotenuse c’ and that the conclusion is an equation relating the lengths: ‘c 2 = a 2 + b 2 ’. Rate the strength of the assumptions and conclusions Let’s rate the assumptions and conclusions. The assumption is about right-angled triangles. Certainly, there are many examples of these, but they are only a small subset of all triangles. Thus we may be tempted to say that this is quite weak but not too weak. But consider this. For any triangle we can produce two right-angled triangles; see Figure 19.1. Thus, despite initial impressions, this theorem will tell us something about all triangles.

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  Journey into Mathematics

  An Introduction to Proofs

  • Joseph J. Rotman(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  legs . In Theorem 2.3, we proved the special case of the Pythagorean theorem involving an isosceles right triangle.

  Figure 2.17

  Theorem 2.6 (Pythagorean Theorem) . In a right triangle with legs of lengths a and b and hypotenuse of length c, we have c 2 = a 2 + b 2 .

   

  Remark . Although the statement of this theorem was accepted centuries before him, Pythagoras was perhaps the first to prove it. The proof we give, due to Indian mathematicians around 400 AD, is based on Figure 2.18 .

  Figure 2.18

  Proof . Figure 2.18 pictures the area of the big square in two ways: first, as a square with side lengths a + b ; second, as dissected into a rhombus PQRS with side lengths c, and four congruent right triangles of area ab . We claim that the rhombus is actually a square. Consider the interior angle γ at P , for example. Note that α + γ + β = 180°. Inasmuch as α + β = 90°, because α and β are the acute angles in a right triangle, we have γ = 90°. The Pythagorean theorem now follows from the algebraic identity

  for the left side is a 2 + 2ab + b 2 , while the right side is c 2 + 2ab .

    The converse of the Pythagorean theorem is also true.  

  Theorem 2.7 . A triangle having sides of lengths a , b , and c with a 2 + b 2 = c 2 must be a right triangle.

   

  Proof . Take two perpendicular lines, as in Figure 2.19 , and choose points A and B with ∣AO ∣ = a and ∣BO ∣ = b . Now ∆OAB is a right triangle having side lengths a, b , and d . The Pythagorean theorem gives a 2 + b 2 = d 2 , and so d = c . But ∆OAB and the given triangle are congruent, by “side-side-side,” and so ∆ is a right triangle.

  Figure 2.19

   

  Figure 2.20 gives another proof, also in the Indian style, of the Pythagorean theorem.

  Figure 2.20

  The square of side length c is partitioned, with a square of side length a – b in the center. This yields

  for the total area. The Pythagorean theorem follows. Query: What does Figure 2.20

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  Geometry and Symmetry

  • L. Christine Kinsey, Teresa E. Moore, Efstratios Prassidis(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Note that sometimes instead of giving the four vertices of a quadrilateral, he will merely describe it as a square or parallelogram and then give two diagonal vertices. This is sufficient to make it clear which figure he is talking about. The side of the triangle subtended by the right angle is called the hypotenuse. Proposition I.47. [Pythagorean Theorem] In a right triangle, the square on the side subtending the right angle is equal to the sum of the squares on the sides containing the right angle. L K N G F H E D C A L N G F H K E D A B B C Proof: Let ABC be a right-angled triangle with the angle BAC right; I say that the square on BC is equal to the squares on BA, AC. 2.5. THE PYTHAGOREAN THEOREM • 65 For let there be described on BC the square BDEC, and on BA, AC the squares GB, HC [Prop. I.46]; through A let AL be drawn parallel to either BD or CE, and let AD, FC be joined. Then, since each of the angles BAC, BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines AC, AG not lying on the same side make the adjacent angles equal to two right angles; therefore CA is in a straight line with AG. [Prop. I.14] For the same reason BA is also in a straight line with AH. And, since the angle DBC is equal to the angle FBA: for each is right: let the angle ABC be added to each; therefore the whole angle DBA is equal to the whole angle FBC. [C.N. 2] And, since DB is equal to BC, and FB to BA, the two sides AB, BD are equal to the two sides FB, BC respectively, and the angle ABD is equal to the angle FBC; therefore the base AD is equal to the base FC, and the triangle ABD is equal to the triangle FBC. [SAS] Now the parallelogram BL is double of the triangle ABD, for they have the same base BD and are in the same parallels BD, AL. [Prop. I.41] And the square GB is double of the triangle FBC, for again they have the same base FB and are in the same parallels FB, GC.

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  Dr. Math Presents More Geometry

  Learning Geometry is Easy! Just Ask Dr. Math

  • (Author)
  • 2005(Publication Date)
  • Jossey-Bass

    (Publisher)

  The first theorems you prove from the postulates are very simple statements. Using them, you prove more and more complicated the- orems, including the Pythagorean theorem (the sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse) and the one that says the sum of the meas- ures of the interior angles in a triangle is 180 degrees. One confusing thing about geometry proofs is that there is no single correct answer. Any logical sequence of true statements that ends with the desired theorem is a valid proof. There are always many correct proofs of any theorem. Some are shorter and simpler than others, and those are usually preferred by teachers, students, and mathematicians. Some are clever, some are tedious, and some are even considered beautiful or elegant. (I’ll bet you didn’t think that aesthetics could enter into mathematics, but it definitely does!) Often proofs are constructed by working backward. Starting with the desired conclusion t, you could say, “If I could prove statement a, then using previously proved theorem (or postulate) b, I could con- clude that t is true.” This reduces your proof to proving statement a, then saying at the end of that proof, “Using theorem b, t is true.” Often there are many possibilities for a (and b). The trick is to pick one that you can prove! Often several plausible choices for a (and b) are tried to find one that works (for you). Logic and Proof 35 Some useful hints: Carefully review the hypothesis and conclu- sion of the theorem you want to prove. Keep in mind the postulates and previously proved theorems that might apply to the situation at hand. Consider working backward, as described earlier. Draw a fig- ure (or several) to illustrate the situation. Consider constructing use- ful lines, points, circles, and so on.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The Pythagorean Theorem and its converse were proved. We discussed the 30uni00B0-60uni00B0-90uni00B0 triangle, the 45uni00B0-45uni00B0-90uni00B0 triangle, and other special right triangles whose lengths of sides determine Pythagorean triples. The final section developed the con- cept known as “segments divided proportionally.” A Look Ahead to Chapter 6 In the next chapter, we will begin our work with the circle. Seg- ments and lines of the circle will be defined, as will special angles in a circle. Several theorems dealing with the measurements of these angles and line segments will be proved. Our work with con- structions will enable us to deal with the locus of points and the concurrence of lines that are found in Chapter 7. Key Concepts 5.1 Ratio • Rate • Proportion • Extremes • Means • Means- Extremes Property • Geometric Mean • Extended Ratio • Extended Proportion 5.2 Similar Polygons • Congruent Polygons • Corresponding Vertices, Angles, and Sides 5.3 AAA • AA • CSSTP • CASTC • SAS~ • SSS~ 5.4 Pythagorean Theorem • Converse of Pythagorean Theorem • Pythagorean Triple 5.5 The 45uni00B0-45uni00B0-90uni00B0 Triangle • The 30uni00B0-60uni00B0-90uni00B0 Triangle 5.6 Segments Divided Proportionally • The Angle-Bisector Theorem • Ceva’s Theorem Summary ■ Summary 271 Methods of Proving Triangles Similar (nABC = nDEF) Figure (Note marks) Method Steps Needed in Proof A C B D F E AA uni2220A _ uni2220D; uni2220C _ uni2220F A C B D F E SSS, AB DE 5 AC DF 5 BC EF 5 k (k is a constant.) Overview ◆ Chapter 5 (Continued)

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  The R Student Companion

  • Brian Dennis(Author)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  143 9 Trigonometric Functions Triangles are an abstract mathematical invention and have been used by human beings for many millennia (Figure 9.1). The mathematical properties of triangles have played a crucial role in human scientific and technological development, from the building of the pyramids to the modern understand-ing of the general theory of relativity and the universe. Any three points not on a line determine a triangle, and triangles within their triangular essence have a large number of shapes. A general triangle has three sides and three interior angles. Geometry tells us that for every triangle the angular measures of the three angles always add up to 1 80 ° . Right Triangles One type of triangle has been singled out for its sheer usefulness. A triangle with one of its angles measuring 90 ° is a “right triangle.” The angular mea-sures of the other two angles of a right triangle must therefore add to 90 ° . The 90 ° angle of a right triangle is usually identified in pictures by a little box (Figure 9.2). One of the immediately useful properties of a right triangle is given by the Pythagorean theorem. If the side opposite to the right angle, called the hypotenuse, has length r (such as in the large triangle in Figure 9.2) and the other two sides have lengths x and y , then for any right triangle the lengths are related as follows: r x y 2 2 2 = + . The Pythagorean relationship can be used to calculate the length of the third side if lengths of the other two sides are known. Another useful feature is that if two right triangles have an additional angle with the same measure, then they are “similar” triangles: Their side lengths have the same proportions. For example, for the two triangles shown in Figure 9.2 all the corresponding ratios of side lengths are equal: y x w v / / = , x r v q / / = , and so on.

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  Course in Triangle Geometry, A

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  • For each natural number n , there exist n Pythagorean triples with different hypotenuses and the same area. • For each natural number n , there exist at least n different Pythagorean triples with the same leg a , where a is some natural number • For each natural number n , there exist at least n different triangles with the same hypotenuse. • In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers. (Actually the radius of the incircle can be shown to be r = n ( m − n )) • The perimeter of a primitive Pythagorean triangle is equal to the sum of the radii of its incircle and the three excircles. • There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. ________________________ WORLD TECHNOLOGIES ________________________ Some relationships Right Triangle with inscribed circle of radius r The radius, r , of the inscribed circle can be found by: The unknown sides of a triple can be calculated directly from the radius of the incircle, r , and the value of a single known side, b . k = b − 2 r a = 2 r + (2 r 2 / k ) c = a + k = 2 r + (2 r 2 / k ) + k The solution to the 'Incircle' problem shows that, for any circle whose radius is a whole number r, setting k = 1 , we are guaranteed at least one right angled triangle containing this circle as its inscribed circle where the lengths of the sides of the triangle are a primitive Pythagorean triple: a =2 r + 2 r 2 b =2 r + 1 c =2 r + 2 r 2 + 1 The perimeter P and area L of the right triangle corresponding to a primitive Pythagorean triple triangle are ________________________ WORLD TECHNOLOGIES ________________________ P = a + b + c = 2 m ( m + n ) L = ab /2 = mn ( m 2 − n 2 ) Additional relationships: If two numbers of a triple are known, the third can be found using the Pythagorean theorem.

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  Diophantine Analysis & Important Mathematical Concepts

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  Generalization: For every integer k > 0, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by 2 k 2 . • If j and k are odd positive integers, not necessarily unequal, there is exactly one primitive Pythagorean triple with a + j 2 = c = b + 2 k . • The hypotenuse of every primitive Pythagorean triangle exceeds the even leg by the square of an odd integer j , and exceeds the odd leg by twice the square of an integer k > 0, from which it follows that: • There are no primitive Pythagorean triples in which the hypotenuse and a leg differ by a prime number greater than 2. • For each natural number n , there exist n Pythagorean triples with different hypotenuses and the same area. • For each natural number n , there exist at least n different Pythagorean triples with the same leg a , where a is some natural number • For each natural number n , there exist at least n different triangles with the same hypotenuse. • In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers. (Actually the radius of the incircle can be shown to be r = n ( m − n ) ) • The perimeter of a primitive Pythagorean triangle is equal to the sum of the radii of its incircle and the three excircles. • There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. ________________________ WORLD TECHNOLOGIES ________________________ Some relationships Right Triangle with inscribed circle of radius r The radius, r , of the inscribed circle can be found by: The unknown sides of a triple can be calculated directly from the radius of the incircle, r , and the value of a single known side, b .

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  Mathematical Techniques in GIS

  • Peter Dale(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  81 5 Plane and Spherical Trigonometry 5.1 BASIC TRIGONOMETRIC FUNCTIONS Trigonometry is concerned with ratios between the sides and angles of triangles. Although at its simplest it is concerned with right-angled triangles on a plane sur-face, its applications extend to many areas of geomatics and geographic information systems (GIS) including calculations on curved surfaces such as that of the Earth. Trigonometry is used extensively in surveying and navigation. In Chapter 3, Figure 3.7 showed similar triangles that have a common shape but differ in scale. In Figure 5.1, the triangles ABC , AB ′ C ′ , and AB ″ C ″ are all the same shape (they have the same angles) but they differ in size or scale. In these triangles the ratio = = = BC AB B C AB B C AB “side opposite over side adjacent.” Given the fixed angle A , then in any right-angled triangle, the ratio BC/AB is constant. This is called the tangent of angle A or tan A . Similarly, the ratio BC/AC is constant = “side opposite over hypotenuse.” This is called the sine of angle A or sin A . Likewise, the ratio AB/AC is a constant = “side adjacent over hypotenuse.” It is called the cosine of angle A or cos A . (See Box 5.1 and Example 5.1.) The ratios sin, cos, and tan are not independent. Given the theorem by Pythagoras AB 2 + BC 2 = AC 2 Dividing both sides by AC 2 we obtain ( AB/AC ) 2 + ( BC/AC ) 2 = 1, or (sin A ) 2 + (cos A ) 2 = 1. Also, = = = A A AB AC BC AC AB AC A (sin / cos ) t an Hence, = tan sin cos It is sometimes helpful to deal with the reciprocal ratios AC/AB and AC/BC . These (and the reciprocal of tan A ) have special names. 1 divided by sine is called cosecant or cosec ; 1 divided by cosine is the secant or sec ; and 1 divided by tangent = cotangent or cot . 82 Mathematical Techniques in GIS By convention, just as x * x = x 2 , so sin A * sin A = (sin A ) 2 and is written as sin 2 A .

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