2 3. Chapters 1-4 of Geometry: Our Cultural Heritage, [26].4. Chapters I-IV of Mathematics in Western Culture, [29].5. Chapters 1 and 2 of The Non-Euclidean Revolution, [44].6. Chapters 1 and 2 of A History of Mathematics, [27].EXERCISES 1.61. A quadrilateral is a four-sided figure in the plane. Consider a quadrilateral whosesuccessive sides have lengths a, b, c, and d. Ancient Egyptian geometers used theformulaA =(а + с)(b + d)EXERCISES 1.6 11to calculate the area of a quadrilateral. Check that this formula gives the correct answerfor rectangles but not for parallelograms.2. An ancient Egyptian document, known as the Rhind papyrus, suggests that the area ofa circle can be determined by finding the area of a square whose side has length thediameter of the circle. What value of is implied by this formula? How close is it tothe correct value?3. The familiar Pythagorean Theorem states that if AABC is a right triangle with rightangle at vertex C and a, b, and c, are the lengths of the sides opposite vertices A, B,and C, respectively, then a² + b² = c². Ancient proofs of the theorem were based ondiagrams like those in Figure 1.6. Explain how the two diagrams together can be usedto provide a proof for the theorem.abВaCCbAaaFIGURE 1.6: Proof of The Pythagorean Theorem4. A Pythagorean triple is a triple (a, b, c) of positive integers such that a² + b² = c².A Pythagorean triple (a, b, c) is primitive if a, b, and c have no common factor. Thetablet Plimpton 322 indicates that the ancient Babylonians discovered the followingmethod for generating all primitive Pythagorean triples. Start with relatively prime(i.e., no common factors) positive integers u and v, u > v, and then define a = u? - v²,b = 2uv, and c = u? + v².(a) Verify that (a, b, c) is a Pythagorean triple.(b) Verify that a, b, and c are all even if u and v are both odd.(c) Verify that (a, b, c ) is a primitive Pythagorean triple in case one of u and v is evenand the other is odd.Every Pythagorean triple (a, b, c) with b even is generated by this Babylonian process.The proof of that fact is significantly more difficult than the exercises above but can befound in most modern number theory books.5. The ancient Egyptians had a well-known interest in pyramids. According to the Moscowpapyrus, they developed the following formula for the volume of a truncated pyramidwith square base:v = c?+ ab + b²).In this formula, the base of the pyramid is an a × a square, the top is a b × b square andthe height of the truncated pyramid (measured perpendicular to the base) is h. One factyou learned in high school geometry is that that volume of a pyramid is one-third thearea of the base times the height. Use that fact along with some high school geometryand algebra to verify that the Egyptian formula is exactly correct.12 Chapter 1Prologue: Euclid's Elements6. Explain how to complete the following constructions using only compass and straight-edge. (You probably learned to do this in high school.)(a) Given a line segment AB, construct the perpendicular bisector of AB.(b) Given a line l and a point P not on l, construct a line through P that is perpendicularto l.(c) Given an angle ZBAC, construct the angle bisector.7. Can you prove the following assertions using only Euclid's postulates and commonnotions? Explain your answer.(a) Every line has at least two points lying on it.(b) For every line there is at least one point that does not lie on the line.(c) For every pair of points A + B, there is only one line that passes through A and B.8. Find the first of Euclid's proofs in which he makes use of his Fifth Postulate.9. A rhombus is a quadrilateral in which all four sides have equal lengths. The diagonalsnra the lina caamanto ininina nnnosita rnara loa the firat fiva Deonositiona of Rook

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to calculate the area of a quadrilateral. Check that this formula gives the correct answer for rectangles but not for parallelograms. 2. An ancient Egyptian document, known as the Rhind papyrus, suggests that the area of a circle can be determined by finding the area of a square whose side has length ° the diameter of the circle. What value of ë is implied by this formula? How close is it to the correct value?