were to be made, which box would hold the most sand? Would one box require more tin than another? Why? Can you draw pictures of the boxes? What is the geometrical name for the sides of the boxes? When boxes are made with triangular bottoms, what shape is usually chosen? Do you understand why? SECTION XV. EQUIVALENT FIGURES. 399. The principle of 393 enables you to mould triangles and all plane figures of geometry almost as easily as if the figures were made of elastic or of clay. pose that two triangles, ▲ A B C and For instance, sup- are of the same height, and are standing, side by side, on the same base line as in Fig. 60. They can be moulded FIG. 60. into one triangle, A C D, with perfect ease. why BCD is equivalent to ABED? Do you see NOTE. In this figure and in future figures, when figures are moulded, the lines along which vertices are moved will be indicated by light dotted lines, as all help lines should be indicated; the resulting lines of the remodelled figure will be indicated by double dotted lines. In your drawings follow the same practice, unless you are able to use different colors for the different kinds of lines. 400. Add together two triangles which have the same altitude, but different bases. Repeat with three pairs of triangles. 401. Can you add three triangles that have the same altitude? Can you do this problem with only one double dotted line? 402. Find an isosceles triangle equivalent to three triangles that have the same altitude. 403. Mould three triangles having the same altitude into a right triangle. E.. D 404. Make six equilateral triangles from one point, O, forming the six-sided figure, or hexagon, represented in Fig. 61. With With a sharp knife cut the figure along the lines O A, O B, etc., beginning at O and not cutting quite up to points A, B, C, etc., except at point F. Place the figure thus F formed so that A, B, C, D, E, F will form a straight line. Can you form one. triangle as large as the hexagon? How long will its base be? How tall will it be? A FIG. 61. B 405. Could you mould a square into a triangle by a process similar to that of 404 ? a parallelogram? The 406. Make a quadrilateral, ABCD, Fig. 62. problem is to mould this quadrilateral into an equivalent triangle. The principle of 393 suggests a method. Divide A B C D into two triangles by the diagonal B D. C D B can now be looked at as a triangle standing on the base D B with vertex at C. C can be moved parallel to D B as far as you wish to move it, and D C and B C will follow the moving just as the legs of the elastic triangle in 391 followed the moving vertex. When C reaches R, in the line of A B, C D B has been moulded into R D B, and the quadrilateral A B C D has been moulded into AARD. This process can be illustrated very prettily on a mouldingtray with clay or sand in the shape of a quadrilateral, and should be so illustrated if possible. 407. If you had moved C in a line antiparallel to D B, where would you have stopped the movement? Illustrate with a figure, using the same quadrilateral as in 406. 408. Draw the diagonal C A, Fig. 62, and mould the quadrilateral into a triangle in two ways, first moving D symparallel with C A, and in a second figure moving D antiparallel to C A. 409. Make a quadrilateral, and mould it into an isosceles triangle. 410. Make a quadrilateral, and mould it into a right triangle. 411. Can you reverse the process of 406, and mould a triangle A D R (Fig. 62) into a quadrilateral A B CD? Suggestion Choose any point between A and R for B, and draw D B as a guide line, to which you must make R C parallel. If all the lines and points of Fig. 62 were blotted out except A RD, what would be your chances of reproducing quadrilateral A B C D? 412. Make a AABC and mould it into a trapezoid. What difficulty can you see in moulding a triangle into a parallelogram by this process? The 413. In moulding a triangle, A B C, into a quadrilateral (Fig. 63), if, after choosing the point P between A and B for one vertex of your quadrilateral, you should move B parallel to P C, not stopping until you should reach AC extended, the figure formed would be a new triangle, A P Q. base of A A B C has been shortened and the altitude lengthened. Hence it is possible to shorten the base of a triangle to any desired length. Illustrate this article with a clay or sand triangle, pushing A FIG. 63. in the base to any desired point and noticing the increase in altitude. 414. In 408 you changed one quadrilateral, A B C D, Fig. 62, into four triangles by using different guide lines, D B and C A, and by moving C and D first in one direction and then in the opposite direction. Place these four triangles on one line, side by side, and shorten all the bases to one length, three-fourths of the line A B. If your work is accurately done, where should the vertices of the raised triangles be? Why? Take pains to have this construction an almost perfect specimen of workmanship. 415. In 413 you shortened the base line just as much or as little as you pleased, but the vertex rose to a point that was out of your control. You, of course, knew that the more you shortened the base the higher the vertex rose, but you could not fix the level to which the vertex would rise, at least without further construction or arithmetical calculation. Can you raise a triangle to any desired altitude? Try to solve the problem by yourselves, but, if you find it too hard, study Fig. 64, where ▲ A B C is the original triangle and H P is the desired altitude. M B is the guide line, parallel to which N H FIG. 64. M M. If QC is the guide to B M in one construction, it is not hard to see that M B becomes the guide to C Q in the reverse construction.. | 416. Make three different triangles and raise the vertex to a level fixed upon beforehand. 417. In 415 we shortened the base by cutting off something at the right end, moving B in as far as Q. Try to solve the same problem by cutting off a length at the left end, thus pushing A in toward B. Suggestion: Extend BC until it crosses P M at N, Fig. 64, and draw a new guide line, N A. Compare the resulting base line with AQ of Fig. 64. 418. Make three triangles, and raise the vertices to anydesired level by pushing the left end of the base toward the right end. Practise changing triangles to triangles of greater altitude A until it is a matter of indiffer ence to you on which side of C B FIG. 65. the triangle you draw your construction lines. 419. Can you now take the necessary steps to lower the |