XIII. The Pythagorean property

The ancient Egyptians discovered that in stretching ropes of lengths 3 units, 4 units and 5 units as shown below, the angle for­med by the shorter ropes is a right angle. 2. The Greeks succeeded in finding other sets of three numbers which gave right triangles and were able to tell without drawing the tri­angles which ones should be right triang­les, their method being as follows. 3. If you look at the illustration you will see a tri­angle with a dashed interior. 4. Each side of it is used as the side of a square. 5. Count the number of small triangular regions in the interior of each square. 6. How does the number of small triangular regions in the two smaller squares compare with the number of triangular regions in the largest square? 7. The Gre­ek philosopher and mathematician Pythagoras noticed the relationship and is credited with the proof of this property known as the Pytha­gorean Theorem or the Pythagorean Property. 8. Each side of a right triangle being used as a side of a square, the sum of the areas of the two smaller squares is the same as the area of the largest square.

 

 

 

Proof of the Pythagorean Theorem

9. We should like to show that the Pythagorean Property is true for all right triangles, there being several proofs of this property. 10. Let us discuss one of them. 11. Before giving the proof let us state the Pythagorean Property in mathematical language. 12. In the triangle above, c represents the measure of the hypotenuse, and a and b represent the measures of the other two sides.

 

 

13. If we con­struct squares on the three sides of the triangle, the area-measure will be a², b² and c². 14. Then the Pythagorean Property could be stated as follows: c² = a² + b². 15. This proof will involve working with areas. 16. To prove that c² = a² + b² for the triangle above, construct two squares each side of which has a measure a + b asshown above. 17. Separate the first of the two squares into two squares and two rectangles as shown. 18. Its total area is the sum of the areas of the two squares and the two rectangles.

A = a²+2ab+b²

19. In the second of the two squares construct four right trian­gles. 20. Are they congruent? 21. Each of the four triangles being congruent to the original triangle, the hypotenuse has a measure c. 22. It can be shown that PQRS is a square, and its area is c². 23. The total area of the second square is the sum of the areas of the four triangles and the square PQRS. A = c²+4 (½ ab). The two squares being congruent to begin with², their area measures are the same. 25. Hence we may conclude the following:

a²+2ab+b² = с²+4(½ ab)

(a² + b²) + 2ab = c² + 2ab

26. By subtracting 2ab from both area measures we obtain a²+ b² = c² which proves the Pythagorean Property for all right trian­gles.

XIV. SQUARE ROOT

 

1. It is not particularly useful to know the areas of the squares on the sides of a right triangle, but the Pythagorean Property is very useful if we can use it to find the length of a side of a triangle. 2. When the Pythagorean Property is expressed in the form c² = a² + b ², we can replace any two of the letters with the measures of two sides of a right triangle. 3. The resulting equation can then be sol­ved to find the measure of the third side of the triangle. 4. For example, suppose the measures of the shorter sides of a right trian­gle are 3 units and 4 units and we wish to find the mea­sure of the longer side. 5. The Pythagorean Property could be used as shown below:

c² = a²+b², c² = 3²+4², c² = 9+16, c² = 25.

6. You will know the number represented by c if you can find a num­ber which, when used as a factor twice, gives a product of 25. 7. Of course, 5x5 = 25, so c = 5 and 5 is called the positive square root (корень) of 25. 8. If a number is a product of two equal factors, then either (любой) of the equal factors is called a square root of the number. 9. When we say that y is the square root of K we merely (вcero лишь) mean that y² = K. 10. For example, 2 is a square root of 4 because 2² = 4. 11. The product of two negative numbers being a positive number, —2 is_also a square root of 4 because (—2)² = 4. The following symbol √ called a radical sign is used to denote thepositive square root of a number. 13. That is √ K means the positive square root of K. 14. Therefore √4 =2 and √25 = 5. 15. But suppose you wish to find the √20. 16. There is no integer whose square is 20, which is obvious from the following computation. 4²= 16 so √16 = 4; a ² = 20 so 4< a <5, 5² = 25, so √25 = 5. 17. √20 is greater than 4 but less than 5. 18. You might try to get a closer approxi­mation of √20 by squaring some numbers between 4 and 5. 19. Since √20 is about as near to 4² as¹ to 5², suppose we square 4.4 and 4.5.

4.4²= 19.36 a² = 20 4.5² = 20.25

20. Since 19.36<20<20.25 we know that 4.4< a <4_:5. 21. 20 being nearer to 20.25 than to 19.36, we might guess that √20 is nearer to 4.5 than to 4.4. 22. Of course, in order to make sure² that √20 = 4.5, to the nearest tenth, you might select values between 4.4 and 4.5, squ­are them, and check the results. 23. You could continue the process indefinitely and never get the exact value of 20. 24. As a matter of fact, √20 represents an irrational number which can only be expressed approximately as rational number. 25. Therefore we say that √20 = 4.5 approximately (to the nearest tenth).

 

APPENDIX

SAMPLE TEST FROM GMAT

1. A trip takes 6 hours to complete. After traveling ¼ of an hour, 1⅜ hours, and 2⅓ hours, how much time does one need to complete the trip?

(A) 2 ¹/¹² hours

(B) 2 hours, 2½ minutes

(C) 2 hours, 5 minutes

(D) 2⅛ hours

(E) 2 hours, 7½ minutes

 

2. It takes 30 days to fill laboratory dish with bacteria. If the size of the bacteria doubles each day, how long did it take for the bacteria to fill one half of the dish?

(A) 10 days

(B) 15 days

(C) 24 days

(D) 29 days

(E) 29.5 days

 

3. A car wash can wash 8 cars in 18 minutes. At this rate how many cars can the car wash wash in 3 hours?

(A) 13

(B) 40.5

(C) 80

(D) 125

(E) 405

 

4. If the ratio of the areas of 2 squares is 2: 1, then the ratio of the perimeters of the squares is

(A) 1: 2

(B) 1: √2

(C) √2: 1

(D) 2: 1

(E) 4: 1

 

5. There are three types of tickets available for a concert: orchestra, which cost $12 each; balcony, which cost $9 each; and box, which cost $25 each. There were P orchestra tickets, B balcony tickets, and R box tickets sold for the concert. Which of the following expressions gives the percentage of the ticket proceeds due to the sale of orchestra tickets?

 

 

P

(A) 100 x ¾¾¾¾¾

(P+B+R)

P

( B ) 100 x ¾¾¾¾¾¾¾¾¾

(12P + 9B + 25 R)

P

( C ) ¾¾¾¾¾¾¾¾¾

(12P + 9B + 25 R)

(9B + 25R)

( D ) 100 x ¾¾¾¾¾¾¾¾¾

(12P + 9B + 25 R)

(12P + 9B +25R)

( E ) 100 x ¾¾¾¾¾¾¾¾¾

(12P)

6. City B is 5 miles east of City A. City C is 10 miles southeast of City B. Which of the following is the closest to the distance from City A to City C?

(A) 11 miles

(B) 12 miles

(C) 13 miles

(D) 14 miles

(E) 15 miles

 

7. If 3 x – 2 y = 8, then 4 y – 6 x is:

(A) -16

(B) -8

(C) 8

(D) 16

(E) cannot be determined

 

8. It costs 10c. a kilometer to fly and 12c. a kilometer to drive. If you travel 200 kilometers, flying x kilometers of the distance and driving the rest, then the cost of the trip in dollars is:

(A) 20

(B) 24

(C) 24 – 2 x

(D) 24 – 0.02 x

(E) 2.400 – 2 x

 

9. If the area of a square increases by 69%, then the side of the square increases by:

(A) 13%

(B) 30%

(C) 39%

(D) 69%

(E) 130%

 

10. There are 30 socks in a drawer. 60% of the socks are red and rest are blue. What is the minimum number of socks that must be taken from the drawer without looking in order to be certain that at least two blue socks have been chosen?

(A) 2

(B) 3

(C) 14

(D) 16

(E) 20

 

11. How many squares with sides ½ inch long are needed to cover a rectangle that is 4 feet long and 6 feet wide?

(A) 24

(B) 96

(C) 3,456

(D) 13,824

(E) 14,266

 

12. In a group of people solicited by a charity, 30% contributed $40 each, 45 % contributed $20 each, and the rest contributed $12 each. What percentage of the total contributed came from people who gave $40?

(A) 25%

(B) 30%

(C) 40%

(D) 45%

(E) 50%

 

13. A trapezoid ABCD is formed by adding the isosceles right triangle BCE with base 5 inches to the rectangle ABED where DE is t inches. What is the area of the trapezoid in square inches?

(A) 5t + 12.5

(B) 5t + 25

(C) 2.5t + 12.5

(D) (t + 5)²

(E) t² + 25

 

 

14. A manufacturer of jam wants to make a profit of $75 by selling 300 jars of jam. It costs 65c. each to make the first 100 jars of jam and 55c. each to make each jar after the first 100. What price should be charged for the 300 jars of jam?

(A) $75

(B) $175

(C) $225

(D) $240

(E) $250

 

15. A car traveled 75% of the way from town A to town B by traveling for T hours at an average speed of V mph. The car travels at an average speed of S mph for the remaining part of the trip. Which of the following expressions represents the time the car traveled at S mph?

(A) VT/ S

(B) VS/4T

(C) 4VT/3S

(D) 3S/VT

(E) VT/3S

 

16. A company makes a profit of 7% selling goods which cost $2,000; it also makes a profit of 6% selling a machine that cost the company $5,000. How much total profit did the company make on both transactions?

(A) $300

(B) $400

(C) $420

(D) $440

(E) $490

 

17. The ratio of chickens to pigs to horses on a farm can be expressed as the triple ratio 20: 4: 6. If there are 120 chickens on the farm, then the number of horses on the farm is

(A) 4

(B) 6

(C) 24

(D) 36

(E) 60

 

18. If x² - y² = 15 and x + y =3, then x – y is

(A) – 3

(B) 0

(C) 3

(D) 5

(E) cannot be determined

 

19. What is the area of the shaded region? The radius of the outer is a and the radius of each of the circles inside the large circle is a/3.

(A) 0

(B) (⅓)pa²

(C) (⅔)pa²

(D) (⁷/₉)pa²

(E) (⁸/₉)pa²

 

20. If 2x – y = 4, then 6x – 3y is

(A) 4

(B) 6

(C) 8

(D) 10

(E) 12

 

21. A warehouse has 20 packers. Each packer can load ⅛ of a box in 9 minutes. How many boxes can be loaded in 1 ½ hours by all 20 packers?

(A) 1 ¼

(B) 10 ¼

(C) 12 ½

(D) 20

(E) 25

 

22. In Motor City 90% of the population own a car, 15 % own a motorcycle, and everybody owns one or the other or both. What is the percentage of motorcycle owners who own cars?

(A) 5 %

(B) 15 %

(C) 33⅓ %

(D) 50 %

(E) 90 %

 

23. Towns A and C are connected by a straight highway which is 60 miles long.

The straight-line distance between town A and town B is 50 miles, and the straight-line distance from town B to town C is 50 miles. How many miles is it from town B to the point on the highway connecting towns A and C which is closest to town B?

(A) 30

(B) 40

(C) 30√2

(D) 50

(E) 60

 

24. A chair originally cost $ 50.00. The chair was offered for sale at 108% of its cost. After a week the price was discounted 10% and the chair was sold. The chair was sold for

(A) $45.00

(B) $48.60

(C) $49.00

(D) $49.50

(E) $54.00

 

25. A worker is paid x dollars for the first 8 hours he works each day. He is paid y dollars per hour for each hour he works in excess of 8 hours. During one week he works 8 hours on Monday, 11 hours on Tuesday, 9 hours on Wednesday, 10 hours on Thursday, and 9 hours on Friday. What is his average daily wage in dollars for the five-day week?

(A) x + (7/5) y

(B) 2 x + y

(C) (5 x + 8 y)/ 5

(D) 8 x + (7/5) y

(E) 5 x +7 y

 

26. A club has 8 male and 8 female members. The club is choosing a committee of 6 members. The committee must have 3 male and 3 female members. How many different committees can be chosen?

(A) 112,896

(B) 3,136

(C) 720

(D) 112

(E) 9

 

27. A motorcycle costs $ 2,500 when it is brand new. At the end of each year it is worth ⁴/₅ of what it was at the beginning of the year. What is the motorcycle worth when it is three years old?

(A) $1,000

(B) $1,200

(C) $1,280

(D) $1,340

(E) $1,430

 

28. If x + 2 y = 2 x + y, then x – y is equal to

(A) 0

(B) 2

(C) 4

(D) 5

(E) cannot be determined

 

29. Mary, John, and Karen ate lunch together. Karen’s meal cost 50% more than John’s meal and Mary’s meal cost ⁵/₆ as much as Karen’s meal. If Mary paid $2 more than John, how much was the total that the three of them paid?

(A) $2833

(B) $30.00

(C) $35.00

(A) $37.50

(B) $40.00

 

30. If the angles of triangle are in the ratio 1: 2:2, then the triangle

(A) is isosceles

(B) is obtuse

(C) is a right triangle

(D) is equilateral

(E) has one angle greater than 80º

 

31. Successive discounts of 20% and 15% are equal to a single discount of

(A) 30%

(B) 32%

(C) 34%

(D) 35%

(E) 36%

 

32. It takes Eric 20 minutes to inspect a car. Jane only needs 18 minutes to inspect a car. If they both start inspecting cars at 8:00 a.m., what is the first time they will finish inspecting a car at the same time?

(A) 9:30 a.m.

(B) 9:42 a.m.

(C) 10:00 a.m.

(D) 11:00 a.m.

(E) 2:00 p.m.

 

33. If x/y = 4 and y is not 0, what percentage (to the nearest percent) of x is

2 x – y

(A) 25

(B) 57

(C) 75

(D) 175

(E) 200

 

34. If x > 2 and y > - 1, then

(A) xy > - 2

(B) – x < 2 y

(C) xy < - 2

(D) – x > 2 y

(E) x < 2 y

 

35. If x = y =2 z and x ● y● z = 256, then x equals

(A) 2

(B) 2 ³√2

(C) 4

(D) 4 ³√2

(E) 8

 

36. If the side of square increases by 40%, then the area of the square increases by

(A) 16%

(B) 40%

(C) 96%

(D) 116%

(E) 140%

 

37. If 28 cans of soda cost $21.00, then 7 cans of soda should cost

(A) $5.25

(B) $5.50

(C) $6.40

(D) $7.00

(E) $10.50

 

38. If the product of 3 consecutive integers is 210, then the sum of the two smaller integers is

(A) 5

(B) 11

(C) 12

(D) 13

(E) 18

 

39. Both circles have radius 4 and the area enclosed by both circles is 28º. What is the area of the shaded region?

 

 

 

(A) 0

(B) 2p

(C) 4p

(D) 4p²

(E) 16p

 

40. If a job takes 12 workers 4 hours to complete, how long should it take 15 workers to complete the job?

(A) 2 hr 40 min

(B) 3 hr

(C) 3 hr 12 min

(D) 3 hr 24 min

(E) 3 hr 30 min

 

41. If a rectangle has length L and the width is one half of the length, then the area of the rectangle is

(A) L

(B) L²

(C) ½ L²

(D) ¼ L²

(E) 2L

 

42. What is the next number in the arithmetic progression 2, 5, 8.......?

(A) 7

(B) 9

(C) 10

(D) 11

(E) 12

 

43. The sum of the three digits a, b, and c is 12. What is the largest three-digit number that can be formed using each of the digits exactly once?

(A) 921

(B) 930

(C) 999

(D) 1,092

(E) 1,200

 

44. What is the farthest distance between two points on a cylinder of height 8 and the radius 8?

(A) 8 √2

(B) 8 √3

(C) 16

(D) 8 √5

(E) 8 (2p + 1)

 

45. For which values of x is x² - 5x + 6 negative?

(A) x < 0

(B) 0 < x < 2

(C) 2 < x < 3

(D) 3 < x < 6

(E) x > 6

 

46. A plane flying north at 500 mph passes over a city at 12 noon. A plane flying east at the same time altitude passes over the same city at 12:30p.m. The plane is flying east at 400 mph. To the nearest hundred miles, how far apart are the two planes at 2 p.m.?

(A) 600 miles

(B) 1,000 miles

(C) 1,100 miles

(D) 1,200 miles

(E) 1,300 miles

 

47. A manufacturer of boxes wants to make a profit of x dollars. When she sells 5,000 boxes it costs 5¢ a box to make the first 1,000 boxes and then it costs y ¢ a box to make the remaining 4,000 boxes. What price in dollars should she charge for the 5,000 boxes?

(A) 5,000 + 1,000 y

(B) 5,000 + 1,000 y + 100 x

(C) 50 + 10 y + x

(D) 5,000 + 4,000 y + x

(E) 50 +40 y + x

 

48. An angle of x degrees has the property that its complement is equal to ¹/ ₆ of its supplement where x is

(A) 30

(B) 45

(C) 60

(D) 63

(E) 72

 

49. The angles of a triangle are in the ratio 2: 3: 4. The largest angle in the triangle is

(A) 30º

(B) 40º

(C) 70º

(D) 75º

(E) 80º

 

50. If x < y, y < z, and z > w, which of the following statements is always true?

(A) x > w

(B) x < z

(C) y = w

(D) y > w

(E) x < w

 

51 ABCD has area equal to 28. BC is parallel to AD. BA is perpendicular to AD. If BC is 6 and AD is 8, then what is CD?

 

(A) 2√2

(B) 2√3

(C) 4

(D) 2√5

(E) 6

 

52. Write formulas according to descriptions:

1. a plus b over a minus b is equal to c plus d over c minus d.

2. a cubed is equal to the logarithm of d to the base c.

3. a) j of z is equal to b, square brackets, parenthesis, z divided by с sub m plus 2. close parenthesis, to the power m over m minus 1, minus 1, close square brackets;

b) j of z is equal to b multiplied by the whole quantity: the quantity two plus z over с sub m, to the power m over m minus 1, minus1.

4. the absolute value of the quantity j sub j of t one, minus j sub j of t two, is less than or equal to the absolute value of the quantity M of t₁ minus b over j, minus M of t₂ minus b over j.

5. R is equal to the maximum over j of the sum from i equals one to i equals n of the modulus of aij of t, where t lies in the closed interval a b and where j runs from one to n.

6. the limit as n becomes infinite of the integral of f of s and j n of s plus delta n of s, with respect to s, from t to t, is equal to the integral of f of s and j of s, with respect to s, from t to t.

7. y sub n minus r sub s plus 1 of t is equal to p sub n minus r sub s plus 1, times e to the power t times l sub q plus s.

8. L sub n adjoint of g is equal to minus 1 to the n, times the n -th derivative of a sub zero conjugate times g, plus, minus one to the n minus 1, times the n minus first derivative of a sub one conjugate times g, plus... plus a sub n conjugate times g.

9. the partial derivative of F of lambda sub i of t and t, with respect to lambda, multiplied by lambda sub i prime of t, plus the partial derivative of F with arguments lambda sub i of t and t, with respect to t, is equal to 0.

10. the second derivative of y with respect to s. plus y, times the quantity 1 plus b of s, is equal to zero.

11. f of z is equal to j sub mk hat, plus big 0 of one over the absolute value of z, as absolute z becomes infinite, with the argument of z equal to gamma.

12. D sub n minus 1 prime of x is equal to the product from s equal to zero to n of, paranthesis, 1 minus x sub s squared, close paranthesis, to the power epsilon minus 1.

13. K of t and x is equal to one over two pi, times the integral of K of t and z, over w minus w of x, with respect to w along curve of the modulus of w minus one half, is equal to rho.

14. The second partial (derivative) of u with respect to t, plus a to the fourth power, times the Laplacian of the Laplacian of u, is equal to zero, where a is positive.

15. D sub k of x is equal to one over two pi, times integral from c minus i infinity to c plus i infinity of dzeta to the k of, w, x to the w divided by w, with respect to w, where c is greater than 1.

 

ACTIVE VOCABULARY

Предлагаемый словарь содержит выражения, которые были представлены в данном пособии как новая лексика. Слова и выражения расположены в алфавитном порядке.

 

-А-

 

абсолютная величина, абсолютное значение	absolute value
абсолютный, полный квадрат	perfect square
абсцисса	abscissa
алгебра	algebra
аксиома	axiom
аксиома завершенности	completeness axiom
аксиома поля	field axiom
аксиома порядка	order axiom
алгебраическая кривая	algebraic curve
анализ	analysis
антилогарифм	antilogarithm
аргумент, независимая переменная	argument
арифметика	arithmetic
арка, дуга	arc
апофема	apothem
ассоциативный	associative

 

-Б-

 

безопасный	secure
бесконечный(о)	infinite(ly)
бесконечно малое приращение	increment
бесконечный предел	infinite limit
бесконечная производная	infinite derivative
бесконечный ряд	infinite series
боковой, латераль­ный	lateral

 

-В-

 

вводить	to introduce
величина, значение	value
вертикальный	vertical
вершина (вершины)	vertex (verti­ces)
ветвь (у гиперболы)	branch
внешний (угол)	exterior
вносить вклад	to contribute
внутренние точки	interior points
внутренний (угол)	interior
вогнутый	concave
воображаемый	fictitious
вписанный круг	inscribed circle
вращение	rotation
выбирать	to select
выбор	choice
выводить, получать, извлекать (о знании)	to derive
выдающийся	distinguished
выпуклый	convex
вырожденный	degenerating
вырождаться	to degenerate
высота под накло­ном	slant height
высота треугольника	altitude
вычисление	computation
вычислять	compute

 

-Г-

 

геометрическое место точек	locus (pl. loci)
горизонтальный	horizontal
гнуть, сгибать, изгибать	to bend (bent-bent)
грань	face
грань, фаска, ребро	edge
градиент	gradient
график	graph

 

-Д-

 

двучленный, биноминальный	binomial
действовать	to operate
действительное число	геаl number
делать вывод	to conclude
делимое	dividend
делитель	divisor
десятичный логарифм	соmmоn logarithm
детерминант, определитель	determinant
директриса	directrix (pl. Dirextrices)
дискриминант	discriminant
дистрибутивный	distributive
дифференциальное исчисление	differential calculus
дифференцирование	differentiation
додекаэдр, двенадцати­гранник	dodecahedron
дробь	fraction
доказывать	to prove
доказательство	proof
дуга, арка	arc

 

-Е-

 

Евклидова геометрия

Euclidean geometry

 

-И-

 

идентичный	identical
изгиб, наклон	slope
измерение, мера, предел, степень	measure
изнурение, истощение, исчерпание	exhaustion
изображение, образ, отраже­ние	image
изобретать	to invent
икосаэдр, двадцатигран­ник	icosahedron
икосидодекаэдр, тридцатидвухгранник	icosidodecahedron
интеграл	integral
интегральное исчисление	integral calculation
интегрирование	integration
интервал	interval
интерпретация	interpetation
иррациональный	irrational
иррациональность	irrationality
исчисление	calculus

 

-З-

 

замкнутый, закрытый	closed
закрытая кривая	closed curve
закрытый интервал	close interval
зеркальное отражение	mirror image
знаменатель	denominator
значительный	significant

-К-

 

касательная	tangent
касательная плоскость	tangent plane
касаться	to concern
кратное число	multiple
квадрант, четверть круга	quadrant
квадратный (об уравнениях)	quadratic
коммутативный	commutative
комплексное	complex
комплексное число	complex nuber
комплектовать	to complete
конгруэнтный	congruent
конечный	finite
конический	conic
конфигурация, очертание	configuration
копланарный, расположеннный в одной плоскости	coplanar
кривая	curve
кривизна	curvature
круглый, круговой	circular
кубическое	cubic
кубооктаэдр, трехгран­ник	cuboctahedron

 

-Л-

 

линейный	linear
линия отсчета	reference line
логарифм	logarithm

 

-М-

 

мантисса	mantissa
математический	mathematical
мгновенный, моментальный	instantaneous
метод бесконечно малых величин	infinitesimal method
многогранник	polyhedron
многогранный, полиэд­рический	polyhedral
многочленный	polynomial
многоугольник	polygon
множество	set
множитель, фактор	factor
момент инерции	moment оf inertia

-Н-

 

наклонная линия, косая линия	oblique
направление	direction
направленные числа	directed numbers
натуральный логарифм	natural logarithm
начало координат	origin
начальная ось	initial axis
независимый	independent
неизменный	unvarying
непрерывная, функция	continuous function
неопределенный	undefined
неуловимый	elusive
нулевой угол	null angle

 

-О-

 

обобщать	to generalize
обозначать	to denote
обозревать	to review
общее значение	total
общий	in commоn
образующая по­верхности	generator - generatrix
обратная величина	reciprocal
обратно	conversely
объем	volume
одновременный	simultaneous
однозначное соответствие, отображение	one-to-one mapping
однообразный	uniform
октаэдр, восьмигранник	octahedron
определять	to determine
ордината	ordinate
основной, главный	principal
ось	axis
открытый	open
открытая кривая	open curve
отношение	relation
отражать	to ret1ect
отрезок	segment
отрицательный	negative
очевидный	obvious

 

 

-П-

 

параллелограмм	parallelogram
пентаграмма	pentagram
переменная величина, функция	fluent
пересекаться	to intersect
перпендикулярный	perpendicular
пирамида	pyramid
Платонов, относящийся к Платону	Platonic
плоскостная кривая	plane curve
плоскостной, плоский	planar
плоскость, плоскостной	plane
площадь всей поверхности	total surface area
поверхность	surface
подмножество	subset
подразделяться, распадаться на	to fall (fell,fallen) into
подразумевать	to imply
подчиняться правилам (законам)	to оbеу laws
познакомиться с	to bе familiar (with)
полный угол	round angle
положительный	positive
полуправильный	semiregular
понятие	notion
понятие, концепт	concept
по часовой стрелке	clockwise
против часовой стрелки	anticlockwise
правильный (о многоугольни­ках и т.д.)	regular
предел отношения	limit оf а ratio
предельный случай	limiting case
предполагать	to assume, to suppose
представлять	to imagine, tо represent
преобразование	translation
призма	prism
применение	application
приписывать	to credit
проекция	projection
произведение	product
производная	derivative
производная, флюксия	fluxion
простой	simple
простое (число)	prime
противоречить	to contradict
противоречие	contradiction
процедура	procedure
прямой	straight
пятиугольник	pentagon
пятиугольный	pentagonal

 

-Р-

 

равносторонний	equilateral
равноугольный	equiangular
радиус	radius
развернутый угол	flat angle
развитие	development
разлагать	to resolve
разложение множителей	factorization
располагаться между	to lie between
рассматривать	to regard
расстояние	distance
рациональный	rational
решать	to solve
рост	growth

 

-С-

 

сводить в таблицу	tabulаtе
свойство	property
сечение	section
система прямоугольных координат	Cartesian coordinates
система записи	notation
скорость, быстрота	velociy
скорость изменения	rate of change
сложение	addition
сокращать	to canсеl
сокращать, преобразовывать	to reduce
соответствовать	tо correspond
средний	mean
средняя величина (значение)	average
ссылаться на	to refer (to)
степень	degree
стереографический	stereographic
сторона (в уравнении)	side
сфера	sphere
сумма	sum
существовать	to exist

-Т-

 

таблица	tablе
твердое тело	solid
тетраэдр, четырехгран­ник	tetrahedron
точка	point
трансцендентный	transcendental
треугольный, трех­сторонний	triangular
трехмерный, объемный, пространственный	three-dimensional

 

-У-

увеличивать	to enlarge
угол понижения (падения)	depression angle
угол возвышения	elevation angle
удлиненный	elongated
удобство	convenience
удовлетворять	to satisfy
указывать	to indicate
умножение	multiplication
уравнение	equation
усекать, обрезать; отсе­кать верхушку	to truncate
ускорение	acceleration
установить	to estabIish

 

-Х-

характеристика

charaсteristic

-Ц-

целый, весь, полный	entire
целое число	integer
цель	purpose
центр массы (тяжести)	centroid

 

 

-Ч-

частное	quotient
часть	unit
числа со знаками	signed numbers
числитель	numerator
числа со знаками	signed numbers

-Ш-

шестиугольник	hexagon
шестиугольный	hexagonal

 

-Э-

 

элемент, составная часть	element
элементарный	elementary
эллипс	ellipse
эллиптический	elliptical