An elementary treatise on geometry : simplified for beginners not versed in algebra . le OPZ, less the triangle DCZ, is to the triangle DCZ,as the hne BK, less the line BG, is to the Une BG;that is, trapezoid DOPC : triangle DCZ = GK : BG;and as GK is (by construction 2) equal to J of AG, trapezoid DOPC : triangle DCZ = J AG : like manner it may be proved that trapezoid DMNC : triangle DCZ = | AG : BG andtrapezoid DABC : triangle DCZ = AG : proportions express that the three trapezoids DOPC,DMNC, DABC, are to each other in the same proportion as onethird is to two thirds to thre
An elementary treatise on geometry : simplified for beginners not versed in algebra . le OPZ, less the triangle DCZ, is to the triangle DCZ,as the hne BK, less the line BG, is to the Une BG;that is, trapezoid DOPC : triangle DCZ = GK : BG;and as GK is (by construction 2) equal to J of AG, trapezoid DOPC : triangle DCZ = J AG : like manner it may be proved that trapezoid DMNC : triangle DCZ = | AG : BG andtrapezoid DABC : triangle DCZ = AG : proportions express that the three trapezoids DOPC,DMNC, DABC, are to each other in the same proportion as onethird is to two thirds to three thirds ; or, which is the same, as oneis to two, to three; whence the rest of the demonstration followsof course. Remark. If it is required to divide the trapezoid ABCD notinto equal parts, but according to a given proportion, it will onlybe necessary to divide the line AG in this proportion, and thenproceed as before. Problem XLI. To divide a given figure into twoparts according to a given proportion^ and in such a wny^that one of the parts may he similar to the whole figure. D. Solution. Let ABCDE be the given figure. 1. Divide one side of the figure, say AB, according toIhe given proportion; let the point of division be Z. 2. Upon AB, as a diameter, describe a semicircle, and 16 182 GEOMETRY- from Z draw the perpendicular ZM, meeting the semi-circle in M. 3. Make Ab =: AM, and upon A6 describe a figure,Abcde, which is similar to the given one, ABCDE (seeProblem XXXIII); the line bcde divides the figure inthe manner required. Demon. The areas of the two similar figures Abcde, ABCDE,are to each other, as the squares upon their corresponding sides(page 98) ; therefore we have the proportion ABCDE : Abcde = AB X AB : A6 X A&.Draw AM and BM ; then AM is a mean proportional betweenAZ and AB ; that is, we have AZ : AM = AM : AB ;and as A6 is, by construction, equal to AM,AZ: A& = A&: AB;consequently the product A& X A6 is equal to AZ X AB. Writing AZ X AB, instead of A& X Ab (its
Size: 1919px × 1303px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookauthorgrundfrancisjfrancisjoseph18051863, bookcentury1800
