Elementary plane geometry : inductive and deductive / by Alfred Baker . metres inlength respectively; show that their areas are as 9 to25, , as (30)2 to (50)2. (For the tliree preceding constructions, the methodof article 4, which follows, should also be employed.) The result of our observations in such cases as thepreceding may be stated thus: Similar triangles are to one another as thesquares of corresponding sides. Note: In the preceding examples it will be ob-served that the lengths of the corresponding sides aresupposed commensurable, , a unit of length can befound that is contain
Elementary plane geometry : inductive and deductive / by Alfred Baker . metres inlength respectively; show that their areas are as 9 to25, , as (30)2 to (50)2. (For the tliree preceding constructions, the methodof article 4, which follows, should also be employed.) The result of our observations in such cases as thepreceding may be stated thus: Similar triangles are to one another as thesquares of corresponding sides. Note: In the preceding examples it will be ob-served that the lengths of the corresponding sides aresupposed commensurable, , a unit of length can befound that is contained in both an exact number oftimes. All lines are not commensurable, though thepreceding statement in black-face is true of aU similartriangles, whether the corresponding sides be commen-surable or not. 4. The following is possibly a more striking way ofpresenting the preceding j^roposition: SiMiLAR Triangles. 143 Let any side, say the base, of a triangle be di\4dedinto as many parts as it contains units of the points of division draw lines parallel to. •i-5-3 S-o the sides, and, through the points of intersection ofthese lines, draw lines parallel to the base. The tri-angle is thus divided into a number of triangles equalto one another in all respects, and all similar to theoriginal triangle. It will be observed that, consideringthese triangles in rows, the rows contain 1, 3, 5, 7, . .triangles, respectively. Hence if the base be 2 unitsin length, the large triangle contains 1 + 3 = 2^ smalltriangles; if 3 units in length, 1 + 3 + 5 = 3^ smalltriangles; if 4 units in length, 1 + 3 + 5-^7 = 42 smalltriangles; and so on. Thus if there be two similartriangles, the base of one containing 3 units of length,and the base of the other 4 units of length, thenumber of small triangles in one will be 3^, and inthe other 4, all such triangles being equal to oneanother. Hence the areas of the triangles are as 3^to 4-, , as the squares of the bases. 144 Geometry. Exercis
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