Chapter 11, Lesson 5 167 What Time is It? To find out what time it is, we can find the angle of elevation of the sun as shown in this figure: sun 6:00 AM tanx=A=i;soa: = 18.435°. 3h 3 The sun's angle of elevation changes at a constant rate because the earth rotates at a constant rate. Let t = the number of hours elapsed 18.435° ^ I 90° ~ 6' t = 1.229 hours = 1 hour, 14 minutes. since 6:00 A.M.: The time was about 7:14 A.M. Chapter 11, Lesson 5 Set I (pages 456-457) In addition to providing practice in using a calculator to find sines and cosines and their inverses, exercises 7 through 12 and the two examples illustrate the fact that a function of an angle is the cofunction of its complement. Exercises 13 through 22 reinforce this idea. The notion of the cosine of an angle developed long after the sine, and the name "cosine" was not introduced until the seventeenth century, the prefix "co-" referring to the word "complement." More information about Bhaskara's calculation of sines can be found in B. L van der Waerden's Geometry and Algebra in Ancient Civilizations (Springer Verlag, 1983). According to van der Waerden, Bhaskara divided an arc of 90° in a circle first into 6, then 12, and finally 24 equal parts. He computed the values of the sines rather than finding them by measurement. Sine and Cosine Practice. , h b • 1. — or . a x + y .2. Aor«. y b .3. y or b x + y 4. — or a 5. 6. a x + y X a h b' •7. 0.996. 8. 0.719. 9. 0.719. •10. 82.935°. •11. 45.000°. 12. 45.000°. Pythagorean Triangle. 13. A right triangle because 5^ + 12^ = 13^. 12 •14. sinA= —. 13 •15. Z A = 67°. 12 16. cosB= —. 17. ZB = 23°. 18. ZA > ZB; ZA and Z B are complementary. 19. sin A = cos B. •20. cos A = — . 13 21. sinB= —. 13 22. cos A = sin B. Finding Lengths and Angles. •23. 5.3. (sin32° =—,x = 10sin32° = 5.3.) 10 24. 18.3. (cos 66° = — , a: = 45 cos 66° = 18.3.) 45 •25. 61°. (tanx= —,a: = 61°.) 9
Transcript
Chapter 11, Lesson 5 167
What Time is It?
To find out what time it is, we can find the angle of elevation of the sun as shown in this figure:
sun
6:00 AM
tanx=A=i;soa: = 18.435°. 3h 3
The sun's angle of elevation changes at a constant rate because the earth rotates at a constant rate. Let t = the number of hours elapsed
18.435° ^ I 90° ~ 6'
t = 1.229 hours = 1 hour, 14 minutes.
since 6:00 A . M . :
The time was about 7:14 A . M .
Chapter 1 1 , Lesson 5
Set I (pages 456-457)
In addition to providing practice in using a calculator to find sines and cosines and their inverses, exercises 7 through 12 and the two examples illustrate the fact that a function of an angle is the cofunction of its complement. Exercises 13 through 22 reinforce this idea. The notion of the cosine of an angle developed long after the sine, and the name "cosine" was not introduced until the seventeenth century, the prefix "co-" referring to the word "complement."
More information about Bhaskara's calculation of sines can be found in B. L van der Waerden's Geometry and Algebra in Ancient Civilizations (Springer Verlag, 1983). According to van der Waerden, Bhaskara divided an arc of 90° in a circle first into 6, then 12, and finally 24 equal parts. He computed the values of the sines rather than finding them by measurement.
•29. BH = 2.6 cm, CI = 5.0 cm, DJ = 7.1 cm, EK = 8.7 cm, and FL = 9.7 cm.
30. sin 30°^ 5^ 10
= 0.50,
sin45°= ^ = 0.71,
sin 60° = — = 0.87, 10
sin 75° = — = 0.97. 10
31. sin 30° = 0.5, sin 45° = 0.7071067 = 0.71, sin 60° = 0.8660254 = 0.87, sin 75° = 0.9659258 = 0.97.
32. It gets larger.
33. A number very dose to zero.
34. A number very close to one.
35. According to a calculator, sin 0° = 0 and sin 90° = 1.
Set 11 (pages 4 5 8 - 4 6 0 )
A wealth of information on flying kites to great altitudes can be found on Richard P. Synergy's Website, www.total.net/~kite/world.html. The line for the kite shown in the photograph was more than 4 miles long and was spooled from a winch. At times it exerted a force of 100 pounds! The calculation of the kite's altitude was more complicated than that implied in the exercises because the line for the kite was not actually straight; its weight made it sag considerably. The altitude of the kite was established through altimeter readings.
As the title of Robert B. Banks' Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics (Princeton University Press, 1998) implies, he includes a good discussion of the mathematics and physics of toppling dominoes. The Guinness Book of World Records always has an entry on the latest record in this sport. In 1988, some students from three universities in the Netherlands spent a month setting up 1,500,000 dominoes. A single push caused 1,382,101 of them to fall!
Kite Flying.
•36. The sine of an acute angle of a right triangle is the ratio of the length of the opposite leg to the length of the hypotenuse.
37. Multiplication.
•38. In a 30°-60° right triangle, the side opposite the 30° angle is half the hypotenuse.
39. In AABC, sin 30° = y . Because sin 30° = 0.5,
y = 0.5, so /z - 0.5/.
•40. Approximately 14,500 ft, or more than
2.7 mi. (sin 4 2 - . ^ ,
14 453 h = 21,600 sin 42° = 14,453; = 2.7.)
5,280
Force Components.
41. /i = 1(27,000) = 13,500 pounds,
V = 13,500 Vs = 23,383 pounds.
42. sin 60° = 27,000'
V = 27,000 sin 60° = 23,383 pounds.
cos 60° = — 27,000
h. = 27,000 cos 60° = 13,500 pounds. Flight Path.
•43. About 806 ft. (sin 3.3° = CB
14,000' CB = 14,000 sin 3.3° = 806.)
44. About 13.6°. (sinA = 3,300 14,000
, Z A = 13.6°.)
Chapter 11, Lesson 6 169
Leaning Tower.
45. About 10 ft. (sin 3° = — , 193
x = 193 sin 3° = 10.1.)
•46. About 4°. (sin A = ZA = 4°.) 193
47. About 18.5 ft. (sin 5.5° =
X = 193 sin 5.5° = 18.5.)
Fiber Optics.
•48. Parallel lines form equal alternate interior angles.
•49. AAS.
50. Corresponding parts of congruent triangles are equal.
•51. 5a.
52. 5d.
53. Because — = — and the cosine of each 5fl d
numbered angle is —. d
SAS Area.
54.
55. s inC = —•,soh = asinC. a
oAABC = hh = ^b{a sin C) = ^ab sin C.
Dominoes.
56. Parallel lines form equal alternate interior angles.
•57. sinZ2 = d-w
58. 30°. {\id-w= hi, sin Zl= 1;
so Z2 = 30° = Z1.)
59. 90°. (If d - w =/i, the domino has room to turn 90° without hitting the next one. Also, iid-w = h, sinZ2 = 1;so Z2 = 90° = Zl.)
60. The next domino might not fall over.
Set 111 (page 4 6 0 ) The first distance to a star beyond the sun was determined by the German astronomer Friedrich Bessel in 1838. The star, 61 Cygni, was chosen because its position against the distant sky changes more than those of most stars. A Scottish astronomer, Thomas Henderson, found the distance to Alpha Centauri (and its faint companion Proxima Centauri) a year later. The measure of the angle ZACS in the figure is called the "parallax" of the star. The unit now used to express distances in space is the "parsec," defined as the distance of a star whose parallax is 1 second. More details on the process of determining astronomical distances by parallax can be found in Eli Maor's book titled Trigonometric DeZ/ghts (Princeton University Press, 1998). Another excellent reference source is Parallax—Ttie Race to Measure the Cosmos, by Alan W. Hirshfeld (W. H. Freeman and Company,
i O O l ) .
Star Distance.
1. About 25,000,000,000,000 mi. ( 0 . 7 6 " = ° : ^ , s i n ° - 7 6 ° - 93,000,000
3,600 3,600 A C
A C = ^ 3 ^ ^ =2.5x10^3) 0.76
sm 3,600 9 5 y 10̂ 3
2. About4.3light-years. (_ _ =4.3.) 5.88 X 10̂ 2
Chapter 1 1 , Lesson 6
Set I (pages 463-465)
About 9 0 % of all avalanches occur on slopes between 30° and 45°, although very wet snow can avalanche on slopes of only 10° to 25°. Skiers sometimes measure the slope angle with a clinometer and sometimes merely estimate it with a ski pole.
According to Cleo Baldon and lb Melchior in Steps and Stairways (Rizzoli, 1989), the eighteenth-century French architect Jacques-Francois Blondel, "concluded that the stride or pace of walking was 24 inches and that when negotiating a stairway this pace should be decreased in
170 Chapter 11, Lesson 6
proportion to the height of the riser. His theory was that two times the height of the riser plus the depth of the tread should equal the 24 inches of the pace." Blondel's formula is still used today, but the pace number has been increased to 26.
Slope Practice.
•1. h.
2. c.
•3. The X-axis.
4. They-axis.
•5. e.
•6. ; , - ; e , - ^ .
7. g.
8. g , 2 ; b , - i .
9. Two nonvertical lines are perpendicular iff the product of their slopes is -1.
10. No. Two lines with the same slope would be parallel. All of the lines named with letters intersect at the origin.
Snow Avalanches.
•11. The slope is tan 30° = 0.58.
12. tan 45° = 1.
Hydraulic Gradient.
•13. -60 m.
14. 1,000 m.
15- -0-06. ( T ^ - ) 1,000
•16. About 3.4°. (tan Z l = 0.06; Z l = 3.4°.)
Stair Design.
17.
18
18
10
10
•18. - (or about 0.56). (^.) 9 18
19. About 29°. (tanA= - , Z A = 29°.)
20.
24
24
21. I (or 0.375). (^.)
22. About 21°. (tan A = - , Z A = 20.6°.) 8
Quadrilateral Problem.
3 •23. •23.
11" 9
24. 7'
3 25. 25.
I T ' 2
26. 3'
• 27. 3 2'
28. Vl30
29. Vl30
30. Vl30
31.
-n
1^^
7
y 10
-11
Chapter 11, Lesson 6 171
32. It is a rhombus because all of its sides are equal. (Or, it is a parallelogram because both pairs of opposite sides are parallel.)
33. They are perpendicular because the diagonals of a rhombus aire perpendicular.
34. They are equal.
35. Their product is -1. (Or, one slope is the opposite of the reciprocal of the other.)
Set 11(pages 465-467)
The two theorems of this lesson about the slopes of parallel and perpendicular lines are only partly proved in exercises 52 through 65. Exercises 52 through 59 establish that, if two nonvertical lines are parallel, their slopes are equal. The converse of this theorem can be proved by showing that
— = — ; so AABC ~ ADEF by SAS similarity, BC EF
from which it follows that Zi = Z 2 , and so || l^. Exercises 60 through 65 establish that, if two
nonvertical lines are perpendicular, the product of their slopes is -1. The converse theorem can be
proved by showing that, if = -1, then AD BD BD BD '
= — . From this it follows that AADB ~ ABDC BD DC
by SAS similarity. Because AADB and ABDC are right triangles, the facts that corresponding angles of similar triangles are equal and the acute angles of a right triangle are complementary can be used to show that ZABC is a right angle; so ± l^.
Slope and x-Intercept.
• 36. Its run is 550 m and its rise is -10 m.
37. 290 m above the ground. (300 -10.)
1 X
38. Its run is a: m and its rise is x (or ) m. 55 55
•39. (300 - — ) m above the ground. 55
•40. 16,500 m. (300 - ~x = 0, —x = 300, 55 55
X = 16,500.)
41. Yes. (16,500 m = 16.5 km > 15 km.)
Point-Slope Equation of a Line.
•42. Run = X - Xj ; rise = y - T/j.
•43. v = mix - X-,) + y,. [m = X-Xi
mix -x^) = y-yyy = mix -x^) + y .̂]
44. y = - — x + 300. ^ 55
45. (Student answer.) (It is hoped, yes.)
46. The points (5,500, 200) and (1,100, 280) are on the path. The point (2,200, 250) is not.
[y = -—(5,500)+ 300 = 200; 55
y = -—(1,100)+ 300 = 280; 55
y = -—(2,200) + 300 = 260 ^ 250.] 55
100-Meter Dash.
•47. From the start to about 2.5 seconds.
48. From about 5 to 8 seconds.
49. From about 2.5 to 5 seconds.
50. The runner's speed is decreasing.
51. The runner's speed is not changing.
Theorem 52. A C
•52.
53.
B C DF EF •
54. Parallel lines form equal corresponding angles.
55. AA.
•56. Corresponding sides of similar triangles are proportional.
57. Multiplication.
•58. Division.
59. Substitution.
Theorem 53.
•60.
61. AD BD"
172 Chapter 11, Lesson 6
•62. The altitude to the hypotenuse of a right triangle is the geometric mean between the segments of the hypotenuse.
63. - 1 rw2
64. Substitution.
65. Multiplication.
Fire Speed.
66. First figure, tan 25° = 0.47; second figure, tan 50° = 1.19.
67. About 6.5 times as fast. The squares of the two slopes are approximately (0.47)̂ = 0.22
and (1.19)2 =1.42; — = 6.5. 0.22
Set III (page467)
The second-century Greek physician Galen first suggested a temperature scale based on ice and boiling water, but not until the seventeenth century were thermometers invented. In 1714, German physicist Gabriel Fahrenheit chose the temperature of a freezing mixture of ammonium chloride, ice, and water for the zero on his scale and the temperature of the human body as 96. The number 96 has many more divisors than 100 does, and Fahrenheit thought it would be more convenient for calculations! Swedish astronomer Anders Celsius introduced his scale, originally called the centigrade scale, in 1742. Strangely, Celsius chose 100° for the freezing point of water and 0° for its boiling point.
Celsius and Fahrenheit.
1. - o r 1.8.
( l = r i ! i = 2 1 2 - 3 2 ^ 1 8 0 ^ 9 ^ run 100 - 0 100 5
2.
3.
F -32 c •
F-32 9
4. C = - ( F - 3 2 ) .
[9C = 5(F-32) ,C= |(F-32).]
5. F = | c + 32. ( F - 3 2 = | c , F = | c + 32.)
5 5 D
6. Yes. The temperature is -40°. If F = C, then
C = | c + 32; so 5C = 9C + 160, - i C = 160, c = ^o .
Chapter 1 1 , Lesson 7
Set I (pages 471-472)
According to David Eugene Smith in his History of Mathematics (G\nn, 1925; Dover, 1958), the Law of Sines "was known to Ptolemy (c.150) in substance, although he expressed it by means of chords." About the Law of Cosines, Smith wrote that it is "essentially a geometric theorem of Euclid" (Book II, Propositions 12 and 13) and that "in that form it was known to all medieval mathematicians."
Greek Equations.
1. Sine.
•2. They are opposite each other.
3
•4. The Law of Sines.
5. Cosine.
6. Z A is opposite side a.
•7. They include ZA.
8. ZB is opposite side p.
9. They include ZB.
10. y^ = a^+ 2a|iaDvr.
11. The Law of Cosines.
Law of Sines.
. sin X sin 75° •12. 40°. (-
smx =
50 75 50 sin 75°
75
•13. 17.0. ( -
= 0.644, X = 40°.)
17 -,x = 17. This is sin 50° sin 50°
consistent with the fact that the triangle is isosceles.)
Chapter 11, Lesson 7 173
sin X sin 14° 14. 75°. (
4 1
sin X = 4 sin 14° = 0.968, x = 75°.)
Law of Cosines. •15. 60°. [192 = 162 + 2l2-2(16)(21)cosx,
361 = 256 + 441 - 672 cos x, 672 cos x = 336, cos X = — = 0.5, X = 60°.]
20. They imply that sin Z l = sin Z3. Substituting r sinZ4 ^ s inZl sinZ3 for , we get = ;
X y y multiplying by y gives sin Z l = sin Z3.
•21. 108°. (Z3 = 1 8 0 ° - Z 2 = 180°-Z1. )
22. sin Z l = sin 72° = 0.9510565 . . .
23. sin Z3 = sin 108° = 0.9510565 . . .
24. They suggest that the sines of supplementary angles are equal.
Set II (pages 4 7 i - 4 7 4 )
In a caption to a figure on which the figure for exercises 25 through 30 is based, John Keay, the author of The Great Arc—The Dramatic Tale of How India Was Mapped and Everest Was Named (Harper Collins, 2 0 0 0 ) , describes the triangulation process: "Given the distance between points A and B (the base-line), the distance from each to point C is calculated by trigonometry using the AB base-line measurement plus the angles CAB and ABC as measured by sighting with a theodolite. The distance between C and B having now been established, it may be used as the base-line for another triangle to plot the position of D.
CD may then be used as the base for a third triangle, and so on." Keay points out that the accuracy of the Survey of India was important not only in measuring a subcontinent but also in determining the curvature of the earth.
As it travels in its orbit around the earth, the moon's distance from the earth varies between 221,463 miles and 252,710 miles. In his fascinating book titled What If the Moon Didn't Exist? (Harper Collins, 1993), astronomer Neil F. Comins observes that, ever since the Apollo 12 astronauts left the laser reflector on the moon in 1969, the distance at any given time can be determined to within a few inches. These measurements reveal that the moon is moving away from the earth at a rate of about 2 inches per year.
27. Z D = 53°. (ZD = 1 8 0 ° - Z 2 - Z 4 = 1 8 0 ° - 4 2 ° - 8 5 ° = 53°.)
•28. DB = 34.6mL (
DB
DB 27.7 sin 85° sin 53°
27.7 sin 85° sin 53°
• 34.6.)
29. CD = 23.2mL ( -CD
CD =
27.7 sin 42° sin 53°
27.7 sin 42° sin 53°
•• 23.2.)
30. C D 2 = C B 2 + D B 2 - 2(CB)(DB)cos 42°, C D 2 = 27.72 + 34.62 - 2(27.7)(34.6) cos 42°, CD^ = 539.96, CD = 23.2.
Distance to the Moon.
31. Z M = 1.4°.
AM 5,820 •32. 238,000 mi. ( -
sin 89.2° sin 1.4°' 5,820 sin 89.20 _
sm 1.4°
174 Chapter 11, Lesson 7
33. 238,000 mi. ( - BM
BM =
5,820 sin 89.4° sin 1.4°'
5,820 sin 89.4° _̂ 238,197.) sin 1.4°
The Case of the Equilateral Triangle.
•34. a'^ = b'^ +c^-2bc cos A.
•35. a^ = a^ + a^- 2aa cos A, a^ = 2a^- 2a^ cos A,
2fl2 cos A = fl2 cos A = i . 2
36. Example figure:
cos 60°= - . 2
The Case of the Right Triangle.
sin A sin B sin C 37.
a 38. 1.
39. ^
b
sinB a b e
•40. By multiplication.
41. The definitions of the sine and cosine based on a right triangle.
42. c2 = fl2 + b 2 _ 2 f l b c O s C .
43. 0.
•44. C2 = fl2 + b2
45. The Pythagorean Theorem.
The Case of the "Flattened" Triangle.
46. c2 = fl2 + b2_2fli,cOsC.
47. -1.
48. C2 = fl2 + b2 + 2flb.
49. Yes. Because c2 = fl2 + 2flb + b2 = (g + i,)2 c = a + b.
Euclid's Law of Cosines.
50. The area of the rectangle is ex. In right
AADC, COS A = --;sox = b cos A. b
Substituting for x, the area of the rectangle is c{b cos A) = be cos A.
Set III (page474)
The idea for this exercise came from Carl Eckart's book titled Our Modern Idol—Mathematical Sc/ence (Scripps Institution of Oceanography, University of California, 1984). The 780-1351-1560 triangle is a remarkably good approximation of a
30°-6o° right triangle because matches VB 780
to five decimal places (^^^ = 1.7320512 . . . and 780
VB = 1.7320508 . . .) and = 2. Using 780
Mathematica reveals that, to four decimal places, AABC is a 30.oooo°-6o.oooo°-90.oooo° triangle. To five decimal places, it is a 30.00000°-6o.oooo3°-89.99997° triangle. To eleven decimal places, ZA = 30.00000000000° but, to twelve decimal places, ZA = 29.999999999996°! Eckart remarks that "such a triangle differs from a right triangle by less than one-tenth of a second" and observes that it cannot be drawn on paper, because "paper is too flexible, and even printed straight lines are too coarse."
A Possible Case of Mistaken Identity.
1. About 30.000000°. [7802 ̂ 13512 + 15602 _ 2(1351)(1560) cos A, Z A = 30.000000°.]
2. About 60.000027°. [By the Law of Sines, sin B _ sin 30° . 1351 sin 30° 1351 " ^ 8 0 " ' " ' ' ' 780 '
ZB = 60.000027° or, by the Law of Cosines, 13512 = 7802 + 15602 _ 2(780)(1560) cos B, ZB = 60.000027°.]
3. AABC is an acute triangle. From the measures of Z A and ZB, it follows that Z C = 89.999973°. Also, 13512 + 7802 = 2,433,601, but 15602 = 2,433,600; so a'^ + b^> c\e triangle is a very good imitation, however, of a 30°-60° right triangle!)
Chapter 11, Review 175
Chapter ^̂ , Review
Set I (pages 476-478)
Exercises 23 through 28 provide another example of a pair of triangles that have five pairs of equal parts and yet are not congruent. Letting AD = 1 and AC = X and using the fact that each leg of a right triangle is the geometric mean between the hypotenuse and its projection on the hypotenuse,
we have '^^^ = —. Solving for x, x̂ = x + 1, X 1
X ^ - X - 1 = O, X = , the golden ratio! In
1 2 AACD, cos A = —; so cos A = —
X 1+
ZA = 51827292 . . a n d ZB = 38.172707 .. °
According to J. V. Field in The Invention of Infinity—Mathematics and Art in the (Renaissance (Oxford University Press, 1997), "the great invention of the military engineers [of the sixteenth century] is the polygonal fort The commonest choice is the pentagon. A construction for a regular pentagon can be found in Elements, Book 4, but it is hardly the sort of construction one would wish to use in laying out the ground plan of a fortress The practical method used by the Venetians, preserved in a file in the State Archives, is shown in [the figure used in the exercises]. We are given the sides of a right-angled triangle. The triangle can be constructed from rope, or by linking together the chains used by surveyors, and repetitions of it will then allow the complete pentagon to be laid out. The person who actually lays out the ground plan using this method does not need to be very good at mathematics, but it is clear that some mathematical expertise was needed to devise the method, and to calculate the lengths of the sides of the triangle."
Greek Cross.
•1. 2.
2. - i 2
3. 2.
4. I •5. - -
7. A C I I D E .
•8. H F I B G .
9. They are equal.
10. Their product is -1. (Or, one slope is the opposite of the reciprocal of the other.)
Doubled Square.
•11. d = sV2.
12. s2.
13. d^.
14. 2s2. [d2 = (sV2)2 = 2s2.]
Escalator Design.
•15. 28 ft. [c = 2fl = 2(14).]
•16. About 24.2 ft. (b = a Vs = 14 Vs = 24.2.)
17. 20 ft. (40 = 2fl.)
18. About 34.6 ft. (fc = a Vs = 20 ^3 = 34.6.)
• 19. About 195 ft. (fc = 338;sofl =
20. About 390 ft. [c = 2fl = 2(195).]
Basketball Angles.
338 •• 195.)
•21. About 5.7°. ^ h 10-7.5 2.5
(tan B = — = = — ^ D 25 25
2.5
= 0.1, ZB = 5.7°.)
22. About 14°. (tan B = ^ = 0.25, ZB = 14°.)
Equal Parts.
23. Three.
24. The altitude to the hypotenuse of a right triangle forms two triangles similar to it and to each other.
25. Three.
•26. Two.
27. Five.
28. No. The longest sides of the two triangles are not equal.
6. A C I C D .
176 Chapter 11, Review
Pentagon.
29. ZBOC = 3 6 ° . ( ^ . )
•30. BC = 1 0 0 f t . ( ^ ^ . )
•31. OB = 137.6 ft. (tan 36° =
OB = 100 tan 36°
137.6.)
32. OC = 170.1 ft. (sin 36° =
OC = 100 sin 36°
100 O B '
100 O C '
: 170.1.)
33. O B 2 + B C 2 = 137.62+ 1002 = 28,934 and OC2 = 170.12 ^ 28,934.
dipping models made from glass or plastic plates into soap solutions!
The Babylonian clay tablet in exercises 65 and 66 was deciphered in 1945. David M. Burton remarks in his History of Mathematics (^AWyn & Bacon, 1985): "This tablet is written in Old Babylonian script, which dates it somewhere between 1900 B.C. and 1600 B.C. The analysis of this extraordinary group of figures establishes beyond any doubt that the so-called Pythagorean theorem was known to Babylonian mathematicians more than a thousand years before Pythagoras was born." Burton observes that the 541-481 error was probably that of a scribe because, in the sexagesimal notation of the Babylonians, 541 is written as 9,1 and 481 as 8,1. Eleanor Robson shows in "Words and Pictures—New Light on Plimpton 322" CAmerican Mathematical Monthly, February 2002) that it is likely that these Pythagorean triples actually arose from computations concerning reciprocals. Even so, "Pythagorean triples" were of interest long before Pythagoras.
Isosceles Right Triangles.
38.
h / >^
/ 2 4
S e t II (pages 4 7 8 - 4 8 0 )
A discussion of various Steiner problems can be found in chapter 3, "Shortest and Quickest Connections," of The Parsimonious Universe-Shape and Form in the Natural World, by Stefan Hildebrandt and Anthony Tromba (Copernicus, 1996). Given a general set of points, the shortest network of segments connecting them is called their "minimal Steiner tree." These trees are important in the design of computer chips and have even been used by biologists studying the relationships of species. A rather amazing natural solution to finding Steiner trees comes from
