Ex. 282. Draw a parallelogram. Through O, the intersection of the diagonals, draw a number of straight lines, meeting the boundary of the parallelogram. Suppose that one of these lines meets the boundary in P and P. Notice that PP' is bisected at O. This is the case for each of the lines. In fact, every straight line drawn through O to meet the boundary in two points is bisected at O. The parallelogram is therefore said to be symmetrical about the point o. O is called the centre of symmetry. Ex. 283. Which of the figures in Ex. 279 are symmetrical about a centre? Ex. 284. Fasten a sheet of paper to the desk (or to a drawing board), and on it draw a parallelogram. Drive a pin through the centre of the parallelogram into the desk. With a knife, cut out the parallelogram. When it is cut free from the sheet of paper, turn it round the pin and see if you can bring it into a position where it exactly fits the hole from which it was cut; what angle must it be turned through to fit in this manner? Ex. 285. Has figure 71 (2) central symmetry? Ex. 286. Describe the symmetry of the following capital letters: A, C, H, I, O, S, X, Z. Solids may have symmetry. The human body is more or less symmetrical about a plane. Consider the reflexion in a mirror of the interior of a room. The objects in the room together with their reflexions form a symmetrical whole; the surface of the mirror is the plane of symmetry. Ex. 287. Give 4 instances of solids possessing planes of symmetry. Ex. 288. Fold a sheet of paper once. Prick a number of holes through the double paper, forming any pattern. On opening the paper you will find that the pin-holes have marked out a symmetrical figure. Join corresponding points as in fig. 72. Notice that when 1 fig. 72. the figure was folded NP' fitted on to NP. This shows that NP' = NP. The line joining any pair of corresponding points, in a figure which is symmetrical about an axis, is bisected by and perpendicular to the axis of symmetry. Ex. 289. If a point P lies on the axis of symmetry, where is the corresponding point P'? Ex. 290. Draw freehand any curve (such as APB in fig. 73); and rule a straight line XY. Mark a number of points on the curve; draw perpendiculars to the line (e.g. PN); produce to an equal distance below the line (e.g. NP' = PN). Draw a curve, freehand, through the points thus obtained. Ex. 291. What points would you describe as "corresponding" in the case of a figure with a centre, but no axis of symmetry? Ex. 292. By a method similar to that of Ex. 290 construct a curve symmetrical about a centre. POINTS, LINES, SURFACES, SOLIDS. This should be taken viva voce; the definitions are not intended to be learnt. In Ex. 109, 116, 210, 221, 224 you have made some solids. The term does not refer to the stuff of which the solids are made, but to the space occupied geometry deals with size and shape, and not with material, colour, hardness, temperature, &c. Any body, such as a brick, a sheet of cardboard or paper, a planet, a drop of water, the water of a lake, the air inside a football, the flame of a candle, a smoke-ring, is called a solid in the geometrical sense of the word. it Ex. 293. Has a brick any length? Has it any breadth? Has any thickness? A solid is bounded by one or more surfaces. Ex. 294. Which of the solids mentioned above is bounded by one surface only? Ex. 295. A bottle is filled partly with water and partly with oil; the water and oil do not mix; the boundary between them is neither water nor oil, it is not a body but a surface. any thickness? Has it Ex. 296. Consider the boundary between the water of a calm lake and the air. Is it water or is it air? Has it any thickness? Has it any length? Has it any breadth ? Ex. 297. Suppose the end of the lake is formed by a wall built up out of the water; what would you call the boundary which separates the wall from the air and water? Has it any thickness? Has it any length? Has it any breadth? A surface has length and breadth, but no thickness. Ex. 298. Part of the surface of the wall is wet and part dry; is the boundary between these two parts wet or dry? Has it any thickness? Has it any length? Has it any breadth? This boundary is really the intersection (or cutting place) of the air-water surface and the wall surface. The intersection of two surfaces is a line. A line has length but no breadth or thickness. We cannot represent a line on paper except by a mark of some breadth; but, in order that a mark may be a good representation of a line, it should be made as narrow as possible. Ex. 299. Take a model of a cube; what are its edges? Have they any length, breadth, or thickness? Ex. 300. If you painted part of your paper black, would the boundary between the black and the white have any width? Ex. 301. If part of the wall in Ex. 297 were painted red and the rest painted black, would the boundary between the two parts be red or black? Ex. 302. Suppose that the red and black paint were continued below the water as well as above, the line bounding the red and black would be partly wet and partly dry; has the boundary between the wet and dry parts of this line any length? The intersection of two lines is a point. A point has neither length, breadth, nor thickness, but it has position. We cannot represent a point on paper except by a mark of some size; the best way to mark a point is to draw two fine lines through the point. We have now considered in turn a solid, a surface, a line, and a point. We can also consider them in the reverse order. A point has position but no magnitude. If a point moves, its path is a line (it is said to generate a line). A pencil point when moved over a sheet of paper leaves a streak behind, showing the line it has generated (of course it is not really a line because it has some thickness). If a line moves, as a rule it generates a surface. A piece of chalk when laid flat on the blackboard and moved sideways leaves a whitened surface behind it. Consider what would have happened if it had moved along its length. If a surface moves, as a rule it generates a solid. The rising surface of water in a dock generates a (geometrical) solid. Ex. 303. Does a flat piece of paper moved along a flat desk generate a solid ? A straight line cannot be defined satisfactorily in a simple way; the idea of a straight line however is familiar to everyone. Ex. 304. How can you roughly test the straightness of (i) a billiard cue, (ii) a railway tunnel, (iii) a metal tube? Ex. 305. How does a gardener obtain a straight line? fig. 74. Ex. 306. Test whether the two thick lines in fig. 74 are straight. |