PROBLEM III. To lay out a given quantity of land in a rectangular form, having the length to the breadth in a given ratio. RULE. As the less number of the given ratio, Is to the greater; So is the given area, To a fourth term.* The square root of this fourth term will be the length required. Having the length, the breadth may be found by the preceding problem. Or it may be found in the same manner as the length. Thus. As the greater number of the given ratio, Is to the less; So is the given area, To a fourth term. The square root of this fourth term will be the breadth required EXAMPLES. 1. It is required to lay out 864 acres in a rectangular form, having the length to the breadth in the ratio of 5 to 3. * DEMONSTRATION. Let ABCD, Fig. 85, be a rectangle, and ABFE and AHGD be squares on the greater and less sides respectively; then (1.6) AD: AE (AB) :: the rectangle AC : square AF. Also AB: AH (AD) :: the rectangle AC: square AG. Hence the As 3:5: 138240; 230400 √230400-480 Perches, the length required. Sq. P. Sq. P. As 5 3 138240:82944 ✓82944-288 Perches, the breadth required 2. It is required to lay out 27 A. 3 R. 20 P. in a rectangular form, having the length to the breadth in the ratio of 9 to 7. 58.897 P. Ans. Length 75.725 P. Breadth PROBLEM IV. To lay out a given quantity of land in a rectangular form, having the length to exceed the breadth by a given difference. RULE. To the given area, add the square of half the given difference of the sides, and extract the square root of the sum; to this root add half the given difference for the greater side, and subtract it therefrom for the less.* * DEMONSTRATION. Let ABCD, Fig. 86, be a rectangle ; in DC let DE be taken equal DA or BC, and let EC be bisected in F ; then (6.2) DF2=DC×DE+FÇ2=DC× AD+FC2=the rectangle AC+the square of half the difference of the sides DC, DA; also DF+FC=DC, the greater side, and DF-FC=DE or DA, the less side. This problem may be neatly constructed thus: take EC equalth given difference of the sides and bisect it in F; make EG' perpen dicular to EC and equal to the square root of the given area, and with the centre F and radius FG describe the arc DG meeting CE produced in D; make DA perpendicular to DC and equal to DE, and complete the rectangle ABCD, which will be the one required. Since (47.1.) FG2=EG+EF2=the given area+the square of half the given difference of the sides, the truth of the construction is plain from the preceding demonstration. EXAMPLES. 1. It is required to lay out 47 A. 2 R. 16 P. in a rectangle, of which the length is to exceed the breadth by 80 perches. 2. It is required to lay out 114 A. 2 R. 35.4 P. in a rectangular form, having the length to exceed the breadth by 15.10 chains. Ans. Length 42.25 Ch. 27.15 Ch. Breadth PROBLEM V. To lay out a given quantity of land in the form of a triangle or parallelogram, one side and an adjacent angle being given. RULE. For a triangle. As the rectangle of the given side and sine of the given angle, Is to twice the given area; So is radius, To the other side adjacent to the given angle. Then having two sides and the included angle given, the other angles and side, if required, may be found by For a parallelogram. As the rectangle of the given side and sine of the given angle, Is to the given area ; So is radius, To the other side adjacent to the given angle.* EXAMPLES. 1. Let AB, BC, Fig. 87, be two sides of a tract of land; the bearing of AB is S. 87° W. dist. 16,25 ch, and the bearing of BC, N, 27o E.; it is required to lay off 10 acres by a straight line AD, running from the point A to the side BC. *DEMONSTRATION. It is demonstrated, prob. 3, sect. 1, Measuring Land, that rad. : sin. B:: ABXBD: 2.ABD (see Fig. 87.); therefore (1.6 cor) rad. XAB: sin B×AB :;AB×BÙ: 2.ABD, or (16.5) sin. BXAB: 2ABD :: rad.XAB: ABXBD :: rad : BD. Since AB DF is equal to 2.ABD, the truth of the rule for the parallelogram is evident. This problem may be constructed as follows; take AB equal the given side and draw BC making the angle B equal the given angle; make BE perpendicular to AB and equal twice the given area of the triangle divided by the given side, or equal the given area of the parallel-ogram divided by the given side; and parallel to AB, draw EF cutting BC in D; join DA, then will ABD be the triangle required, or complete the parallelegram ABDF, for the one required. The reason of the construction is plain. 2. Given the side AB, Fig. 15, of a parallelogram, equal 20 ch. and the angle A 63° 30'; required the side AC, that the content may be 21 As ABxsin. A acres. AB 20 ch. sin. A 63° 30' Ar. Co. 8.69897 0.048 1 2.33.44 10 00000 1.07962 3. Given one side of a triangle, equal 30 perches, anangle adjacent to this side 71° 15', and the area 2 acres; required the other side adjacent to the given angle. Ans. 24.53 perches. 4. Given one side of a parallelogram, equal to 32.26 ch., an angle adjacent to this side 83° 30', and the area 74 acres; required the other side adjacent to the given angle. Ans. 23.09 Ch. PROBLEM VI. The area and base of a triangle being given to cut of a given part of the area by a line running from the angle opposite the base. RULE As the given area of the triangle, To the base corresponding to that area.* |