a rectangle. Draw a figure showing the square and rectangle meant. What is true of their areas? Why? The process described in 438 gives you the means of moulding a square into a rectangle having one side any desired length longer than the side of the square. Do you see how? 443. Turn four squares into rectangles. 444. Turn one square into four different rectangles. R A FIG. 69. 445. In 437 and 438 we have seen how to turn an isosceles right triangle into a right triangle with unequal legs, in two different ways. Can a right triangle with unequal legs be changed to an isosceles right triangle? Try first to reverse the method of 437, and to change a right triangle A S R, Fig. 69, to an isosceles right triangle A B C by lowering the vertex R to the proper level C, so that the extended base A B shall be equal to the reduced height A C. How will you take the first step? Why is it difficult to draw a guide line, parallel to which R may be moved? 446. In the next place, try to reverse the method of 438. To help the imagination, place triangle A S R (Fig. 70) in a position something like that of triangle ASR in Fig. 68. The problem still is to find C, to which to move the vertex S. We know that C is somewhere on a line drawn parallel to RA from S, but we do not know exactly where. It is through the triangle corresponding to ACE (Fig. 68) that we shall be able to find C. In 440 you learned that ▲ ACE is a right triangle. How can you find the point corre FIG. 70. sponding to E? Review 272, and, with the aid of the principle therein stated, find point C. AC will be the side of the isosceles triangle desired. Draw A B at right angles to A C and equal to it. Complete triangle A B C. 447. Change six different right triangles into isosceles right triangles by the method of 446. Vary the position of the triangles, that you may not depend upon having your figure placed as in Fig. 68. 448. Change a rectangle into a square, following the method suggested by 446. Repeat three times. 449. Change a triangle into a square. change the triangle into a rectangle. Suggestion: First 450. Change a pentagon into a square. Make the steps in their proper order perfectly clear. 451. Add a triangle and a quadrilateral, and change the resulting figure into a square. 452. Put three equal squares side by side, so as to form a rectangle. Mould the rectangle thus formed into a square form. 453. Find a square three times as large as a given square. Find a square five times as large as a given square. 454. Cut a square into five equal rectangular strips, finding of one side with your dividing tool. Change one of these strips into a square form. How is this new square related to the original square See 454. See 454. 455. Find a square of a given square. 456. Find a square of a given square. 457. Make a pentagon. Can you find a square that shall be as large as the pentagon? 458. In changing a square into a rectangle, 438, you found that a great many rectangles could be formed equivalent to the square. The shape of the rectangle depends upon the distance which the vertex of the square is moved along C E (see Fig. 68). In changing a rectangle into a square, how many results can you get? 459. Make a square, A B C D, Fig. 71. Change the square into a rectangle eight times by moving each vertex in the two directions indicated by the arrows at the vertex (Fig. 71). Move each vertex the same distance, and compare the resulting rectangles. NOTE. The object of this exercise is to give you practice in changing squares into rectangles with the squares in different positions. If you find any one position hard for you, practise that case until it is as simple as any of the others. 460. Draw a square A B C D (Fig. 72), and change it, by 438, to rectangle K HAF. The FIG. 71. fact that the square is equivalent to the rectangle may be expressed as follows: A D'AFA H. Show that E D'EFX ES = Suggestion: Imagine that D has been moved to A, parallel to N E. 461. What kind of quadrilateral is SH A E, which is made up of the two rectangles SKFE and K HAF? Show that A E2 ~~ A D2 + DE2. 462. Does the truth stated in 461 apply to all right triangles ? 463. The principle of 460 and 461 stated in words is as fol SKFE. FIG. 72. lows: The square formed on the hypotenuse of a right triangle is equivalent to the sum of the squares formed on its sides. This principle is called the Pythagorean principle, be cause the discovery of it is attributed to Pythagoras, a famous Greek mathematician, who lived about 600 B.C., and who did much to encourage the study of Geometry. How Pythagoras made his discovery is not known. The method of showing its truth here given is essentially that devised by Euclid, who lived three hundred years later than Pythagoras, and who was the most famous of the ancient Geometers. 464. The Pythagorean principle makes the addition of squares even simpler than the addition of triangles; for it is necessary merely to place the squares to be added on the legs of an imaginary right triangle, as in Fig. 72. The square on the hypotenuse is the square desired. 465. Add two squares. Repeat the process four times, using different squares. 466. Can you form two unequal squares on one line, A B, as a side? If A B is a given line, can you form A B2 without any further description? What is meant by the statement that a square is determined by one of its sides? 467. Can you find the side of a square equivalent to MN+ST, where M N and S T are given lines, without constructing any square? Describe your process carefully. It is better, in the majority of cases, not to draw the squares actually, but to use the sides of the squares. 468. Given two lines, a and b, find a2 + b2 in a square form. Find a square equivalent to a2 + a2 or 2 a2; to a2 + 2 a2 or 3 a2. Find a square equivalent to 3 a2 by 453, and compare the result with that just obtained. Compare your method of getting the side of 2a2 here with the method suggested in 315. 469. Given three lines a, b, c. Can you find a square equivalent to a2 + b2 + c2? Find a2 + a2+ a2 by this method. Find 3a2 by the methods of 453 and 468, and compare the results with that obtained here. 470. Change two triangles into squares, and add the resulting squares. Add the two triangles without changing them to squares; change the resulting triangle to a square, and compare the results obtained by the two methods. 471. Make a square on a given line, a. Make a square on a line equal to 2a. Compare the two squares, a2 and (2a). How much larger than the first square is the second? 472. Make a square on the line 3a as a base, and compare the resulting square with a2. 473. What is the side of a square equivalent to 16 a2? to 25 a2? to 100 a2? 474. Make a square on the line a. Draw a line M N equal to 3a and, at right angles to M N, draw N P equal to a. Join M P, and call it b. How many times as big as a2 is b2? 475. Can you devise a simple method of finding 5 a2 from the experiment in 474? Find a square five times as large as a2 by the method of 453, and compare the result with that obtained by your new method. 476. Can you draw a right triangle by first drawing the hypotenuse? Recall what you know about the middle point of the hypotenuse. 477. Make a line A B, and draw a semi-circumference, of which A B is the diameter. Join any point, C, of this semi-circumference with A and B. Can you tell the size of BCA? Give a reason for your answer. 478. Make a square, a2, of the 474 and 475. With a line A B equal to 3a as a diameter, describe a semi-circumference. See Fig. 73. Make a line A C, called a chord of the circumference, equal to a. Draw the chord C B. same size as that used in Draw squares on A B, A C, and B C. за FIG. 73. Why? 479. Draw a new semi-circumference of the same size as |