552. Can you make a hexagon of a given hexagon? of a given hexagon? of a given hexagon? In each case the hexagon must be similar to the given hexagon. 553. Can you make a circle equivalent to the sum of two given circles? 554. Can you subtract one circle from another, giving the difference in the form of a circle? 555. Can you change a circular ring, or band, into a circle? 556. Can you make a circle twice as large as another circle? three times as large? four times as large? 557. One circular hole was found to have of the diameter of another; how much smaller than the second hole was the first hole? 558. If you wish to double the amount of light coming through a circular window, should you double the diameter of the window ? H 559. Draw a circumference, Fig. 80, and with your dividing-tool divide the diameter into five equal parts; draw semicircumferences on the diameters A B, A C, A D, and A E above the diameter AF, and semicircumferences on the diameters B F, C F, D F, and EF below the diameter A F. What part of the semicircle A AHF is AKB? What part of the whole circle is semicircle AK B? What part of the whole circle is semicircle BMF? The horn-shaped figure AKBFTA is made up of R P B COD E M T FIG. 80. the semicircles AK B+ ATF - BMF. What part of the whole circle is the horn ? If the corresponding lines of similar figures have the same relation to each other, what part of the semicircum ference A II F should the semicircumference AK B be? What part of the semicircumference A HF should the semicircumference B M F be? Which is the longer path from A to F, the path by the semicircumference A T F, or that by the two semicircumferences A K B and B MF? 560. By adding and subtracting the right semicircles in Fig. 80, the surface of the strip A LCNF MBK A can be found. Can you select the right semicircles, and can you estimate what part of the whole circle each semicircle is, and what part of the whole circle the strip is? Can you estimate the perimeter of the strip? 561. In 559 you saw that the horn A KBFTA is equivalent to the semicircles A KB+ ATF - BMF; can you find one semicircle that shall be equivalent to A K B + ATF - BMF? See 553 and 554. 562. Make a circle and divide the diameter into eight parts; draw semicircumferences, as in Fig. 80, cutting the circle into eight strips. Try to show that the circle is thus divided into eight equivalent strips. 563. Can you find a circle one-half of a given circle? one-third of a given circle? two-fifths of a given circle? 564. Review carefully Sections XV., XVI., and XVII. (in several lessons if necessary). 565. Write a sketch of the square, describing how squares can be added and subtracted; how they can be enlarged any number of times without losing their shape; and how they can be divided into as many small squares as may be desired. Tell, also, what other figures can be added, subtracted, enlarged, or diminished by the methods used for squares. SECTION XVIII. CIRCLES AND INSCRIBED ANGLES. 566. A circle is a figure bounded by a line, called the circumference, every point of which is equally distant from a point within, called the centre. The sign for circle is O, plural; and that for circumference is O, plural R. You have already learned that you can measure an angle by placing its vertex at the centre of a circle whose circumference has been divided into three hundred and sixty equal parts, and by noting the number of these parts, or degrees, between the sides of the angle. Any portion of a circumference is called an arc, and the angle formed by the radii through the ends of an arc is said to subtend the arc. Thus in Fig. 81 A BOA subtends the arc B A, written BA. The number of degrees in the arc B A is the same as the number of degrees in the angle BOA at the centre; BA = 60°, BOA = 60°; if / BA = 50°, / B 0 A = 50° ; if BA = 2°, BOA = 2°; so that an angle at the centre of a circle has the same name as the arc which it subtends. This principle is simply a restatement of the principle of the protractor. FIG. 81. 567. In using a protractor you have always placed the centre of the instrument upon the vertex of the angle. The peculiarities of circles enable us often to measure angles without a protractor, and also to measure angles whose vertices are not at the centre of the circle. In Fig. 82 A O E is a diameter, and AB = 60°. Can you, without a protractor, give the value in degrees of all of the angles in the figure? Can you see any simple relation between BEA and BA? = 120 568. Imagine, or draw, an arc A C 70°, and the diam-, eter A E. Can you tell how many degrees there are in the angle CEA? Try arcs of 90°, 75°, 30°, 120°, and 150°, from A; and, after joining the end of each arc with E, find the value of each angle formed at E with the line E A. Draw a figure like Fig. 82 for each case, making the arc A B the proper length each time. 569. An angle, placed as angle BE A is placed in Fig. FIG. 82. 82, with its vertex on the circumference, and with its sides running to other points of the circumference, forming chords, is called an inscribed angle. The experiments of 567 and 568 suggest the principle that inscribed angles, which have a diameter for one side, subtend arcs of twice their own number of degrees. Try to prove this principle by showing that if E = x°, ÁÈ _must_be equal to 2x Fig. 83. Draw H O and compare Lx. to HA? HOA with How is HO A related 570. What is the largest inscribed angle that you can draw with one side a diameter? From one end of a diameter draw chords that shall make angles of 60°, 45°, 30°, 2210, FIG. 83. A and 75° with the diameter, using your compasses and ruler only. 571. Place the inscribed angle x, so that the centre of the circle shall come between the two chords which form its sides, If be inscribed, as in Fig. 85, in such a way that both chords come on the same side of the centre, will the arc AH still contain twice as many degrees as the angle x? Draw E OK, the diameter, and call AEK, y, as before; HEK = [ x + Ly. KA = 2y°, and HK =2x+ KA = = 2x° as be fore. We therefore can say that any inscribed angle subtends an arc of twice its own number of degrees. H ix x+y τσ FIG. 85. 572. How large an arc must an inscribed angle of 90° subtend? an angle of 221° ? an angle of 150° ? 573. Several arcs of a circumference, when measured, were found to be respectively, 70°, 83°, 90°, 340°, and 180°; how large was each inscribed angle that subtended these arcs? 574. An inscribed angle is said to stand on the arc which it subtends. Can you say why all inscribed angles in one circle which stand on the same arc must be equal? Make |