1. The document discusses geometric concepts such as lines, angles, and the Pythagorean theorem. 2. Equations and formulas are presented for calculating lengths of sides of right triangles based on the Pythagorean theorem. 3. Approximations of irrational numbers like the square root of 2 and pi are calculated through successive decimals.

5. a b
c

6. a, b c
c
a b

8. n

9. c
a b

10. A B C
C
A B

13. The Pythagorean Proposition

17. a, b c a b
c
a b
a, b c a
b c a b °
Euclid's Elements

19. ABC
a,b c a b c
a b
a b
c
- -
a b

20. ,
c

21. • a b c
• a b <c
• a b >c

22. 0, 1, 5,
-7, , , A
B
A
B
1.

23. 2.

24. -
1 5
1.

25. - -
2. ,-
,-

26. - - - - - - - - (1)
1) 10
- - - - - - - - (2)
10
2) 1) - - - - - - - - -
-

27. --------
(1)
1) 1000
- - - - - - - - (2)
1000
1) 10
- - - - - - - - (3)
10
2) 3) - - - - - - - -
--

29. 2
1 2
2 1 4

30. 1 2
1 2
1.1 1.2 1.3 1.4 1.5
1.21 1.44 1.69 1.96 2.25
1.4 1.5
1.4 1.5
1.41 1.42 1.43
1.4 1.51.9881 2.0164 2.0449
1.41
1.42

31. 1.411 1.412 1.413 1.414 1.415
1.990921 1.993744 1.996569 1.999396 2.002225
1.414 1.415
1.4141 1.4142 1.4143
1.99967881 1.99996164 2.00024449
1.4142
1.4143 1.41421 1.41422
1.9999899241 2.0000182048
1.41421 1.41422

32. 1.414213562373095048801688724209...
2

33. 1) 1.234567891011121314
2) 3.4323223222...
3) 16.79779777977779...
4) -4.399339933399...
3.141592653589793238462...
3.1416 3.142

34. 2
- 2
-
1
-
1
2

35. 1.
1
900
900 -

36. 2
784
784 - -
3

37. 2.
15 16
10 9
165 169

38. 1.
3 4
3 4
9 13 16
2. 13 9 16
16
3.5
3.5 3.6 3.7
12.25 12.96 13.00 13.69
3. 13
12.96 13.69
3.60 3.603.61
12.9600 13.0000 13.0321

39. 4. 13
12.9600 13.0321 13.0321
3.605
3.605 3.606
12.996025 13.000000 13.003236
5. 13
13.003236 12.996025
3.606
3.61

40. 3.
1 100
10 3.162 100 10.000 0.1000
11 3.317 121 1.331 0.0909
12 3.464 144 1.728 0.0833
13 3.606 169 2.197 0.0769
14 3.742 196 2.744 0.0714
15 3.873 225 3.375 0.0667
16 4.000 256 4.096 0.0625

41. 3.1
=
x + 2 = 59

42. x
ax + b = 0 a, b
a#0
a=b b=a a b
1. a+b=c c=a=b

43. a=b b=c a=c
a, b c
x=5+7 5 + 7 = 12
x = 12
x = -3y -3y = 0.5
x = 0.5

44. a=b a+c=b+c
a, b c
1. a=5 a+3=5+3
3
-7
2. x+7=2 (x + 7) -7 = 2 -
7

45. a =b ca = cb a,
b c
3
1. m + 1 = 2n
3(m + 1) = 3(2n)
2. -3x = 15 -1
3
-3x = 15-3 -3
-3

46. 2x = 0 1x+ 3 = 0 8y – 11 =
9y + 7
2
2x – 5 = 0

47. 15x + 9 = 69
15x + 9 = 69
9
15x + 9 – 9 = 69 – 9
15x = 60
15
.
. 15x – 15 .
. = 60
– 15
X =

48. 0 1
" "
x y

50. m
b
y x2, y1/3,
xy

52. ( )
x y

53. 2
n
a1,a2,...,an x1,x2,...,xn
b
3
x1,x2,x3
x,y,z
n–1 (hyperplane)
n

54. A B
A, B, C
1
A≥0

55. (simultaneous
equations)
t m = V/T
x (VU−WT) / V y
(WT−VU) / T
T = p−h, U = h, V = q−k, W=k
t 0 (h,
k) 1 (p, q) t 0
1
(interpolation) t

61. 2

63. AB AB
X Y
AB