23 SECOND SECTION. LENGTHS. IV. TABLES OF LINEAL MEASURE. 52. The student is probably already acquainted with the Table of Measures of Length; but for convenience we will give it here: 12 inches make 1 foot. 3 feet make 1 yard. 6 feet make 1 fathom. 16 feet or 5 yards make 1 rod or pole. 40 poles make 1 furlong. 8 furlongs make 1 mile. 53. In measuring land a chain is used, called Gunter's Chain, which is 22 yards long, and consists of 100 equal links; each link is therefore 22 of a yard long, that is 7.92 inches. Thus 25 links make a pole, 10 chains or 1000 links make a furlong, and 80 chains or 8000 links make a mile. V. RIGHT-ANGLED TRIANGLE. 54. When we know the lengths of two of the three straight lines which form a right-angled triangle, we can calculate the length of the third straight line. We shall now give the rules for this purpose, which depend on the theorem of Art. 30, as will be more clearly seen hereafter. 55. The sides of a right-angled triangle being given, to find the hypotenuse. RULE. Add the squares of the sides and extract the square root of the sum. 56. Examples: (1) One side is 8 feet, and the other is 6 feet. The square of 8 is 64, and the square of 6 is 36; the sum of 64 and 36 is 100; the square root of 100 is 10. Thus the hypotenuse is 10 feet. (2) One side is 2 feet, and the other is 10 inches. 2 feet are 24 inches; the square of 24 is 576, and the square of 10 is 100; the sum of 576 and 100 is 676: the square root of 676 is 26. Thus the hypotenuse is 26 inches. 57. In the example just solved one side was given expressed in feet, and the other expressed in inches; before we applied the rule for finding the hypotenuse we turned the feet into inches, so that both the sides might be expressed in the same denomination. In like manner before using any rule in mensuration, it is necessary to express all the given lengths in the same denomination. We may work with all the lengths expressed in inches, or with all expressed in feet, or with all expressed in yards, or with all expressed in any other denomination; but we must not work with some of the lengths expressed in one denomination, and some expressed in another. 58. In the two examples solved in Art. 56, the square root could be found exactly, and so the length of the hypotenuse was determined accurately. But it may happen that the square root cannot be found exactly; in such a case we can continue the process for extracting the square root to as many decimal places as we think necessary. 59. Examples: (1) One side is 3 feet 4 inches, and the other is 2 feet 8 inches. 3 feet 4 inches=40 inches, 2 feet 8 inches=32 inches. Thus if we proceed to two decimal places we find that the hypotenuse is approximately 51.22 inches. (2) One side is 24 feet, and the other is 1.2 yards. Thus if we proceed to two decimal places we find that the hypotenuse is approximately 4:32 feet; or, taking the nearest figure, we may say that it is 4:33 feet. 60. The hypotenuse and one side of a right-angled triangle being given, to find the other side. RULE. From the square of the hypotenuse subtract the square of the given side, and extract the square root of the remainder. Or, Multiply the sum of the hypotenuse and the side by their difference, and extract the square. root of the product. 61. Examples. (1) The hypotenuse is 10 feet, and one side is 8 feet. The square of 10 is 100, and the square of 8 is 64; take 64 from 100, and the remainder is 36; the square root of 36 is 6. Thus the other side is 6 feet. Or thus: the sum of the hypotenuse and the given side is 18; their difference is 2; the product of 18 and 2 is 36: the square root of 36 is 6. (2) The hypotenuse is 26 inches, and one side is 10 inches. The square of 26 is 676, and the square of 10 is 100; take 100 from 676, and the remainder is 576: the square root of 576 is 24. Thus the other side is 24 inches. Or thus: the sum of the hypotenuse and the given side is 36; their difference is 16; the product of 36 and 16 is 576: the square root of 576 is 24. 62. We have given two forms of the Rule in Art. 60; the first form is more obviously connected with the Rule in Art. 55; the second form is generally more convenient in practice, as requiring less work. 63. In the two examples solved in Art. 61 the square root could be found exactly, and so the length of the side was determined accurately. But it may happen that the square root cannot be found exactly; in such a case we can continue the process for extracting the square root to as many decimal places as we think necessary. 64. Examples: (1) The hypotenuse is 1 foot 9 inches, and one side is 14 inches. 1 foot 9 inches = 21 inches. 245-0000 (15.65 21+14=35, 21-14=7, 35 × 7=245. 1 25) 145 125 306) 2000 3125) 16400 15625 775 Thus if we proceed to two decimal places, we find that the required side is approximately 15 65 inches. feet. (2) The hypotenuse is 27 yards, and one side is 3′4 27 yards=81 feet. 81+34=11.5, 54-0500 (7.35 81-34=4'7. 11.5 4.7 805 460 540 5 49 Thus if we proceed to two decimal places, we find that the required side is approximately 735 feet. 65. We will now solve some exercises which depend on the Rules already given. (1) One side of a right-angled triangle is 408 feet; the sum of the hypotenuse and the other side is 578 feet: required the hypotenuse and the other side. By Art. 60 the square of 408 is equal to the product of the sum of the hypotenuse and the other side into their difference; therefore if the square of 408 be divided by 578, the quotient will be the difference of the hypotenuse and the other side. In this way we find that the difference of the hypotenuse and the other side is 288. |