Elements of geometry and trigonometry . 106 GEOMETRY. PROBLEM XV. To construct a figure similar to the figure P, and equivalent tothe figure Q. Find M, the side of a squareequivalent to the figure P, andN, the side of a square equiva-lent to the figure Q. Let X bea fourth proportional to the threegiven lines, M, N, AB ; uponthe side X, homologous to AB,describe a figure similar to the figure P ; it will also be equiva-lent to the figure Q. For, calling Y the figure described upon the side X, we haveP : Y : : AB2 : X2; but by construction, AB : X : : M : N,or AB^ : X- : : M^ : N^ ; hence P : Y
Elements of geometry and trigonometry . 106 GEOMETRY. PROBLEM XV. To construct a figure similar to the figure P, and equivalent tothe figure Q. Find M, the side of a squareequivalent to the figure P, andN, the side of a square equiva-lent to the figure Q. Let X bea fourth proportional to the threegiven lines, M, N, AB ; uponthe side X, homologous to AB,describe a figure similar to the figure P ; it will also be equiva-lent to the figure Q. For, calling Y the figure described upon the side X, we haveP : Y : : AB2 : X2; but by construction, AB : X : : M : N,or AB^ : X- : : M^ : N^ ; hence P : Y : ; M^ : N^. But byconstruction also, M^=P and N2=:Q; thereloioP : Y : : P :Q; consequently Y=Q; hence the figure Y is similar to thefigure P, and equivalent to the figure PROBLEM XVL To construct a rectangle equivalent to a given square, and havingthe sum of its adjacent sides equal to a given line. Let C be the square, and AB equal to the sum of the sidesof the required rectangle. Upon AB as a diame-ter, describe a semicir-cle ; draw the line DEparallel to the diameter,at a distance AD from it,equal to the side of thegiven square C ; from the point E, where the parallel cuts thecircumference, draw EF perpendicular to the diameter ; AFand FB will be the sides of the rectangle required. For their sum is equal to AB ; and their rectangle isequivalent to the square of EF, or to the square of AD ; hencethat rectangle is e(^ivalent to the given square C. Scholium. To render the problem possible, the distance ADmust not exceed the radius ; that is, the side of the square Cmust not exceed the half of the line AB.
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
