side of the hill the angle of depression of the base of the building is observed to be 14° 36′, and the angle of elevation of the top is 21° 43'. A level line from the instrument meets the building 19 ft. 7 inches above the base. Find the height of the building. 12. A balloon is observed, at the moment it passes over a level road, from two points in the road an eighth of a mile apart. The angles of elevation from the two points are 33° 17′ and 42° 6'. Find the distances of the balloon from the two observers. 13. In surveying, it is sometimes desired to extend such a line as AB in the figure beyond an obstacle. Show that this could be done by means of a broken line ABECD. What measurements would be necessary to determine the distance BC and the angle ECD? A B E D 14. How far to the side of a target 1300 ft. away should a gunner aim from a ship going 15 mi. per hour, if the speed of the bullet is 2000 ft. per second and he fires at the instant he is directly opposite? 15. From a railway train going 50 mi. per hour a bullet is fired 1000 ft. per second at an angle of 75° 28'.3 with the track ahead. Find its speed and direction. 16. A man in a railway car going 45 mi. per hour observes the raindrops falling at an angle of 10° with the horizontal. Assuming that the raindrops are actually falling vertically, find their speed. 17. The resultant of two forces is 10 lb. ; one of the forces is 8 lb. and makes an angle of 36° with the resultant. Find the magnitude of the other force. (Two solutions.) 18. A horse pulls a canal boat by a rope which makes an angle of 25° 35' with the tow path. What size of engine would propel the boat at the same speed? (Assume that the horse is doing one "horse power.") 19. A man climbs a hill inclined (on the average) 32° with the horizontal. His pocket barometer shows that at the end of 2 hours he has increased his elevation 2750 ft. Find his average speed up the slope. 20. Find the areas of triangles which have the following given parts: 21. Find the area of a triangular piece of ground having two angles, respectively, 73° 10' and 90° 50', and the side opposite the latter 150.6 rods. CHAPTER IV DIRECTED LINE SEGMENTS AND ANGLES PART I. GENERAL DEFINITIONS AND PRINCIPLES 34. Directed Lines and Segments. Vectors. Such expressions as north and south, right and left, up and down, call attention to the necessity of distinguishing between the two directions of a straight line in order to express our ideas with precision. It is often convenient to select one direction on a straight line as the positive direction; the other is then called the negative direction. Thus, if two forces act along the same line, FIG. 42 but in opposite directions, it is convenient to call one positive and the other negative. A line on which a choice of directions has been made is called a directed line.* In drawings, the positive direction of a directed line is indicated by an arrow head. A portion of a line between two of its points, A, B, is called a segment. We distinguish between the two possible directions of a segment as follows: The notation AB means the segment whose initial point is A and whose terminal point is B, while BA means the segment whose initial point is B and whose terminal point is A. A force is indicated graphically by such a directed line segment, whose direction indicates the direction of the force, and whose length indicates the intensity or amount of the force. Velocities are represented in the same manner. See p. 43. A segment is said to be positive if its direction coincides with the positive direction of the line on which it lies; otherwise it *The assignment of a positive direction on one line does not determine the positive direction on any other line, but it is often convenient in the case of parallel lines to choose the same direction on each as the positive direction. In what follows this choice will be understood unless the contrary is specified. 48 B C D + + + F + GH + FIG. 43 is a negative segment. In Fig. 43, AB, FG, etc., are positive segments; DA and FE are negative. Two segments are said to have the same sense if they lie on the same line or on parallel lines, and if both are positive or both are negative. Two segments are said to be of opposite sense if they lie on the same line or on parallel lines, and if one is positive and the other is negative. Two segments having the same length and the same sense are equal; if they have the same length and are of opposite sense, each is the negative of the other. Thus, in Fig. 43, AB= EF, while AC - GE and CB=— FG. The numerical measure of a directed segment is the number of units in its length with the sign + or —, according as the segment is positive or negative. B C D With the agreements of this article, a directed segment may be substituted for any parallel segment of the same magnitude. Such a movable directed segment is often called a vector. + + B' foot B 35. Geometric Addition of Line Segments. Given two line segments AB and CD, of the same or opposite sense; to add the second to the first, place the initial point of the second on the terminal point of the first; then the segment from the initial point of the first to the terminal point of the second is their sum. Thus, lay off A'B'= AB and B'D' = CD; then A'D' = AB + CD (Fig. 44). This sum in the case of forces or velocities is a special case of the resultant, defined on p. 43, when the forces (or velocities) lie on the same lines. Given 36. Subtraction of Line Segments. two segments AB and CD of the same or opposite sense, to subtract the second from the first, add the negative of the second E C B D B' D' B B' D FIG. 45 to the first; in other words, place the terminal point of the subtrahend on the terminal point of the minuend; then the segment from the initial point of the minuend to the initial point of the subtrahend is the required difference. EXERCISES XVII. ADDITION AND SUBTRACTION OF SEGMENTS 1. By laying them off on a directed line with some convenient unit, add the segments whose numerical measures are 3 and 4; 2 and — 4; - 4 and 5; - 2 and -3. 2. Find the sum, or resultant, of two forces that act in the same line whose intensities (measured in pounds) are 5 and 10, respectively. Draw a figure to represent the solution. 3. If three forces of intensities +7, 15, +2 (lb.), respectively, act on a body in the same line, find the resultant force. Draw a figure. 4. If a man walks with a speed of 4 mi. per hour toward the rear of a train going 35 mi. per hour, find his actual speed. Draw a figure. 5. A man's gains and losses (indicated by —) in business in successive months are $250, $118, $35, $712, - $15. Find the total gain, and the average gain per month. Draw a figure. 6. The gains and losses in the population of a city in successive years are 3500, 1100, 2300, 600, + 2800. Find the total gain, and the average gain per year. 7. Verify that AB + CD = CD + AB by laying off segments. 8. Verify that (AB+ BC) + DE = AB+(BC + DE). [NOTE. A segment whose end points coincide is called a zero segment, or simply, zero.] 10. Verify that if A, B, C, are any three points on a line, then no matter what the order of the points, or which is the positive direction of the line, always AB + BC = AC (six cases). 11. Lay off the segments AB = 8, CD = 10, BC = 6, DE = 5, and perform geometrically the following operations, checking each by the corre 12. By n times a segment, or n AB, we mean the segment where n is a positive integer. If AB= 4, construct 3. AB. *For it is required to find a segment which can be substituted for x in the equation x + CD = AB, and AB+ DC is the unique solution of this equation. . = 13. Construct 5 times a segment whose numerical measure is 3/5. 14. If AB = n CD, we say that CD AB ÷ n. By the well-known method of plane geometry we can find points P1, P2, P3, ......, Pn−1, such that API P1P2 = P2P3: = Pn-1B = AB/n, where n is any positive = ... integer. Draw a segment at random and divide it into fifths. 37. Rotation. Directed Angles. In describing rotation, it is convenient to regard angles as positive or negative in the following manner: an angle is thought of as generated by the rotation of one of its sides about the vertex as center; its first position is called the initial side, the final position is called the terminal side. An angle generated by a rotation opposite to the motion of the hands of a clock (counterclockwise), is said to be positive; an angle generated by a clockwise rotation, is said to be negative.* FIG. 46 Thus, in Fig. 46, α, B, 8, are positive angles; y is negative. Logically, the familiar units of angle used in elementary geometry and thus far in this book may be defined as follows: if the terminal side of the angle rotates in the positive direction until it coincides (for the first time), (a) with the perpendicular to the initial line at the vertex, the angle is a right angle; (b) with the prolongation of the initial side through the vertex, the angle is a straight angle; (c) with the initial line itself, the angle is a revolution (or a perigon). A degree (°) is defined in terms of any one of these by the equation 360° = 1 rev. = 2 str. 4 rt. ; it is divided into minutes (') and seconds ("), so that 1° = 60′ : 3600". An acute angle is a positive angle less than a right angle. An obtuse angle is greater than a right angle and less than a straight angle. Angles may be of any magnitude, positive or negative. Thus, in Fig. 46, for example, ß is greater than a straight angle; and 8 is greater than 360°, or a complete revolution. In rotating parts of machinery, such angles have a very vivid meaning. Thus, a wheel which rotates 370° per second has a very different speed from that of a wheel which rotates 10° per second. * Either of these directions may of course be chosen as the positive direction of rotation, the other is then the negative direction. The choice here made is the customary one for angles; but in many kinds of machinery, the other sense of rotation is considered positive, as in the case of a clock. |