350. To construct a polygon similar to two given similar polygons and equivalent to their difference. Let R and R' be two similar polygons, and AB and A'B' two homologous sides. It is required to construct a similar polygon which shall be equivalent to R' - R. Construct the rt. Z P, and take PO = A B. From O as a centre, with a radius equal to A' B', describe an arc cutting PX at H. On A" B", homologous to AB, construct the polygon R" similar to R. Then R" is the polygon required. RR: A B2 : A B2, $343 (similar polygons are to each other as the squares on their homologous sides). For PROPOSITION XXI. PROBLEM. 351. To construct a triangle equivalent to a given Let ABC DHE be the given polygon. It is required to construct a triangle equivalent to the given polygon. From D draw D E, and from H draw HF || to D E. Produce AE to meet HF at F, and draw D F. The polygon ABC D F has one side less than the polygon ABCDHE, but the two are equivalent. For the part A B C D E is common, and the ▲ DEF: = ADEH, for the base D E is common, and their vertices Fand H are in the line FH to the base, $325 (A having the same base and equal altitudes are equivalent). Again, draw CF, and draw DK to CF to meet A F produced at K. Draw C K. The polygon ABCK has one side less than the polygon. ABCDF, but the two are equivalent. For the part A B C F is common, = and the ACFK: ACFD, for the base CF is common, and their vertices K and D are in the line KD || to the base. $325 In like manner we may continue to reduce the number of sides of the polygon until we obtain the ▲ CIK. Q. E. F. 352. To construct a square which shall have a given ratio to a given square. S It is required to construct a square which shall be to R as n is to m. On a straight line take A B =m, and BC = n. On A C as a diameter, describe a semicircle. At B erect the LBS, and draw SA and SC. Then the AA SC is a rt. A with the rt. at S, § 204 On SA, or SA produced, take SE equal to a side of R. Then SF is a side of the square required. (a straight line drawn through two sides of a ▲, parallel to the third side, divides those sides proportionally). that is, the square having a side equal to SF will have the same ratio to the square R, as n has to m. Q. E. F. 353. To construct a polygon similar to a given polygon and having a given ratio to it. Let R be the given polygon and the given ratio. n m It is required to construct a polygon similar to R, which shall be to R as n is to m. Find a line, A'B', such that the square constructed upon shall be to the square constructed upon A B as n is to m. it § 352 Upon A'B' as a side homologous to AB, construct the polygon S similar to R. (similar polygons are to each other as the squares on their homologous sides). PROPOSITION XXIV. PROBLEM. 354. To construct a square equivalent to a given paral Let ABCD be a parallelogram, b its base, and a its Upon MO as a diameter, describe a semicircle. At N erect NPL to MO. Then the square R, constructed upon a line equal to NP, is equivalent to the ABCD. For MN NP :: NP: NO, § 307 (alet fall from any point of a circumference to the diameter is a mean proportional between the segments of the diameter). .. Np2 = Μ Ν ΧΝΟ = a X b, § 259 (the product of the means is equal to the product of the extremes). Q. E. F. 355. COROLLARY 1. A square may be constructed equivalent to a triangle, by taking for its side a mean proportional between the base and one-half the altitude of the triangle. 356. COR. 2. A square may be constructed equivalent to any polygon, by first reducing the polygon to an equivalent triangle, and then constructing a square equivalent to the triangle. |