CHAPTER XVI. Squares and Circles In and About Circles and Squares. 1. To inscribe a square in a circle, draw two diameters at right angles to one another and join their extremities. The construction being accurately made, the set-square will show that the angles A, B, C, D are all right angles; and the equality of the sides AB, may be proved by BC, using the dividers. D Of course the evident equality of the triangles AOB, BOC, proves the equality of the sides, and the angles ABC, BCD . . . are all right angles, because they are angles in semicircles. Inscribe a square in a circle of radius 40 millimetres. Test the accuracy of your construction by examining, with the dividers, the equality of the sides. Inscribe a rectangle (which is not also a square) in a circle. Test the accuracy of your construction by examining, with the set-square, whether the angles are all right angles. In a circle whose radius is 3 inches, inscribe a rectangle, one of whose sides is 1 inch. With instruments test the success of your construction, the equality of opposite sides, the parallelism of opposite sides, the right-angledness of the figure. E H 2. To describe a square about a given circle, draw two diameters at right angles to each other, and through the ends of each diameter draw lines parallel to the other. The construction being accurately made, the set-square will show that the angles of E, F, G, H are all right angles, and the equality of the sides EF, FG, . . may be proved by using the dividers. A .... 8 Evidently the figures AOCE, AODF, .. are equal squares, whence we readily prove that the sides of EFGH are all equal; and its angles are right angles. Describe a square about a circle whose radius is 30 millimeters. Test the accuracy of your construction by finding whether the sides are equal, using the dividers; and use the set-square to determine whether the angles are right angles. Describe a square about a circle whose radius is 13 inches. As in the previous question, test the accuracy of your construction. Draw two diameters in a circle not at right angles to each other, and draw tangents at their extremities. Determine the nature of the figure formed by the tangents by measuring the lengths of its sides. 3. To inscribe a circle in a given square, draw portions of the diagonals of the square, so that they intersect, as at E. Draw EF perpendicular to one of the sides. With EF as radius, describe a circle. If the construction has been accurate the circle will touch the sides of the square. B By drawing the complete diagonals it may readily be shown, from the equality of such triangles as EFD, EGD, that the perpendiculars from E on the sides are equal. Describe a square with side of 4 inches, and in it inscribe a circle. Show, by measurement with dividers and set-square, that the lines joining the points of contact form a square. Show that the sides of this are perpendicular to the diagonals of the original square. Inscribe a circle in the second square of the preceeding question. Inscribe a circle in a rhombus, each of whose sides. is 4 inches, and one of whose angles is 60°. 4. To describe a circle about a given square, draw portions of the diagonals so that they intersect. Then, placing the sharp point of the compasses at E, where the diagonals intersect, and the pencil point on any one of the angles, and describing a circle, it will pass through the other angular points of the square. X The lines from E to the angles are equal if the square has been accurately constructed and the diagonals accurately drawn; for the diagonals of all parallelograms bisect each other, and the diagonals of a square are equal. Construct a square whose side is 80 millimetres, and about it describe a circle. Construct a square whose side is 40 millimetres, and about it describe a circle. At the angular points of the square in the preceding question draw tangents to the circle, and, by measurement with the dividers and set-square, show that the tangents form a square. About the square formed by the tangents in the preceding question describe a circle. The sides of a rectangle are 80 and 35 millimetres. Describe a circle about it. Starting with a square whose side is 100 millimetres, inscribe a circle in it, then a square within this circle, a circle within the last square, etc. With the angular points of a square as centres, describe four circles, such that each touches two of the others. Describe a circle to touch these four circles. If ABCD be a square, and from AB, BC, CD and DA equal lengths AE, BF, CG, DH be cut, what is the figure EFGH? Exercises. 1. Inscribe a square in a circle of radius in. Test accuracy of construction. 2. Inscribe a square in a circle of radius 11⁄2 in. Test accuracy of construction. Compare length of side of square with that of side of square in previous question. Compare area of square with that of square in previous question. 3. Describe a circle of radius 12 in. In it draw two diameters making an angle of 30° with one another, and join their extremities. What is the resulting quadrilateral? Apply tests. 4. Describe a circle of radius 30 millimetres, and in it construct a rectangle one of whose sides is 25 millimetres. Test accuracy of construction. 5. Describe a circle of radius 60 millimetres, and in it construct a rectangle one of whose sides is 50 millimetres. Test accuracy of con struction. Compare the length of the longer side of this rectangle with the length of the longer side of the rectangle in the preceding question. How are the areas of the rectangles related? 6. Describe a circle of radius in., and about it describe a square. Test accuracy of construction. 7. Describe a circle of radius 35 millimetres, and both in and about it construct squares. 8. What ratio always exists between the sides of squares about and in the same circle? What ratio between their areas? 9. Draw two diameters of a circle (radius 1 in.) at an angle of 30° to one another, and at their ends draw tangents. What is the resulting quadrilateral about the circle? Apply test. Show that 10. About a circle of radius 35 millimetres construct a rhombus with angles 60° and 120°. Test accuracy of construction. the length of each side must be 70 millimetres. 11. Why is it that a rectangle or parallelogram about a circle must always be a square or rhombus ? 12. About a circle of radius 1 in. construct a rhombus with one angle three times the other. What is the length of the sides? |