1. A quadrilateral is a four-sided figure in the plane. Consider a quadrilateral whose successive sides have lengths a, b, c, and d. Ancient Egyptian geometers used the formula A = (а + с)(b + d) EXERCISES 1.6 11 to calculate the area of a quadrilateral. Check that this formula gives the correct answer for rectangles but not for parallelograms.
1. A quadrilateral is a four-sided figure in the plane. Consider a quadrilateral whose successive sides have lengths a, b, c, and d. Ancient Egyptian geometers used the formula A = (а + с)(b + d) EXERCISES 1.6 11 to calculate the area of a quadrilateral. Check that this formula gives the correct answer for rectangles but not for parallelograms.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 1
![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 1.6
1. A quadrilateral is a four-sided figure in the plane. Consider a quadrilateral whose
successive sides have lengths a, b, c, and d. Ancient Egyptian geometers used the
formula
A =
(а + с)(b + d)
EXERCISES 1.6 11
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 is implied by this formula? How close is it to
the correct value?
3. The familiar Pythagorean Theorem states that if AABC is a right triangle with right
angle 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 on
diagrams like those in Figure 1.6. Explain how the two diagrams together can be used
to provide a proof for the theorem.
a
b
В
a
C
C
b
A
a
a
FIGURE 1.6: Proof of The Pythagorean Theorem
4. 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. The
tablet Plimpton 322 indicates that the ancient Babylonians discovered the following
method 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 even
and 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 be
found in most modern number theory books.
5. The ancient Egyptians had a well-known interest in pyramids. According to the Moscow
papyrus, they developed the following formula for the volume of a truncated pyramid
with 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 and
the height of the truncated pyramid (measured perpendicular to the base) is h. One fact
you learned in high school geometry is that that volume of a pyramid is one-third the
area of the base times the height. Use that fact along with some high school geometry
and algebra to verify that the Egyptian formula is exactly correct.
12 Chapter 1
Prologue: Euclid's Elements
6. 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 perpendicular
to l.
(c) Given an angle ZBAC, construct the angle bisector.
7. Can you prove the following assertions using only Euclid's postulates and common
notions? 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 diagonals
nra the lina caamanto ininina nnnosita rnara loa the firat fiva Deonositiona of Rook](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F97b2fd98-425c-4e74-b0df-f3dfeb9a3e94%2Fwfs9epf_processed.png&w=3840&q=75)
Transcribed Image Text: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 1.6
1. A quadrilateral is a four-sided figure in the plane. Consider a quadrilateral whose
successive sides have lengths a, b, c, and d. Ancient Egyptian geometers used the
formula
A =
(а + с)(b + d)
EXERCISES 1.6 11
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 is implied by this formula? How close is it to
the correct value?
3. The familiar Pythagorean Theorem states that if AABC is a right triangle with right
angle 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 on
diagrams like those in Figure 1.6. Explain how the two diagrams together can be used
to provide a proof for the theorem.
a
b
В
a
C
C
b
A
a
a
FIGURE 1.6: Proof of The Pythagorean Theorem
4. 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. The
tablet Plimpton 322 indicates that the ancient Babylonians discovered the following
method 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 even
and 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 be
found in most modern number theory books.
5. The ancient Egyptians had a well-known interest in pyramids. According to the Moscow
papyrus, they developed the following formula for the volume of a truncated pyramid
with 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 and
the height of the truncated pyramid (measured perpendicular to the base) is h. One fact
you learned in high school geometry is that that volume of a pyramid is one-third the
area of the base times the height. Use that fact along with some high school geometry
and algebra to verify that the Egyptian formula is exactly correct.
12 Chapter 1
Prologue: Euclid's Elements
6. 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 perpendicular
to l.
(c) Given an angle ZBAC, construct the angle bisector.
7. Can you prove the following assertions using only Euclid's postulates and common
notions? 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 diagonals
nra the lina caamanto ininina nnnosita rnara loa the firat fiva Deonositiona of Rook
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