117. What should be the diameter of a water pipe that supplies the same amount of water as both an 8-in. and a 6-in. pipe, if we assume that all other conditions, as pressure, etc., are the same in the three pipes? 118. A rail 50 ft. long must be bent through what angle (i.e. 20 formed by perpendiculars at its end) if the radius of curvature equals 360 ft.? R 119. In laying a curved track, a rail 55 ft. long is bent through an angle of 17° 30'. Find the radius of curvature. 120. If O is the center of arc AB, BC 1 OA, and 20 is small, then the difference between AB and BC is very small. If we have to find the value of BC, we may, in such cases, find the length of arc AB instead. If 0=4°, the error due to this simplification equals .00005 of the result, and for smaller angles the error is much smaller, since it is approximately proportional to the cube of the angle. C A If BC is a tower standing on a horizontal plane A O, BO = 2000 ft. and ≤ 0 = 2°. Find the height of the tower. 121. The apparent diameter of the moon equals 30', i.e. a line EM drawn from earth (E) to the center of the moon (M) and a tangent EA include an angle of 15'. If EM=243,000 mi., find the radius of the moon. E M 122. The apparent diameter of the sun is approximately the same as that of the moon (preceding problem). What is the diameter of the sun if its distance from the earth equals 93,000,000 mi.? SOLID GEOMETRY BOOK VI LINES AND PLANES IN SPACE-POLYHEDRAL ANGLES 477. DEF. Geometry of space or solid geometry treats of fig ures whose elements are not all in the same plane. (17) 478. DEF. A plane is a surface such that a straight line joining any two points in it lies entirely in the surface. (14) A plane is determined by given points or lines, if one plane and only one, can be drawn through these points or lines. PROPOSITION I. THEOREM 479. A plane is determined: (1) By a straight line and a point without the line. (2) By three points not in the same straight line. (3) By two intersecting straight lines. (4) By two parallel straight lines. (1) To prove that a plane is determined by a given straight line AB and a given point C. Turn any plane EF passing through AB about AB as an axis until it contains C. If the plane, so obtained, be turned in either direction, it would no longer contain C. E XC B F Hence, the plane is determined by AB and C. (2) To prove that three given points 4, B, and a determine a plane. Draw AB. Then AB and C determine the plane. (Case 1.) (3) To prove that two intersecting straight lines AB and AC determine a plane. [Proof by the student.] (4) To prove that two parallel lines AB and CD determine a plane. The parallel lines AB and CD lie in the same plane by definition. Since AB and the point C determine a plane, the two parallels determine a plane. B A B 480. COR. The intersection of two planes is a straight line. For the intersection cannot contain three points not in a straight line, since only one plane can be passed through three such points. NOTE. Constructions in solid geometry are usually treated as if we possessed the physical means of making a plane that is determined by any of the above elements. Obviously this could be done only if we employed models. With ruler and compasses the work is largely symbolic; i.e. we obtain pictures which are symbols of the figures to be constructed, but we do not always obtain the required figures themselves, as we do in plane geometry. Ex. 1438. Why is a photographer's camera or a surveyor's transit supported by three legs? Ex. 1439. How many planes are determined by four points, which do not all lie in one plane? 481. DEF. A straight line and a plane are parallel if they do not meet, however far they may be produced. 482. DEF. Two planes are parallel if they do not meet, however far they may be produced. PROPOSITION II. THEOREM 483. The intersections of two parallel planes with a third plane are parallel. Given the parallel planes I and II, intersected by a third plane III in AB and CD, respectively. Proof. AB and CD cannot meet, for otherwise planes I and II would meet. Hence AB and CD lie in the same plane. AB CD. Q. E. D. 484. COR. Parallel lines included between parallel planes are equal. NOTE. The student should note that in order to prove the parallelism of two lines in solid geometry it is not sufficient to show that the lines cannot meet. It must be demonstrated also that the two lines lie in one plane. PROPOSITION III. THEOREM 485. A plane containing one, and only one, of two parallel lines is parallel to the other line. A M Given AB || CD, and plane MN containing CD only. Proof. AB and CD determine a plane, intersecting MN in CD. Hence, if AB meets MN, it must meet MN in CD. Hence plane MN | AB. Q. E. D. 486. REMARK. Proposition III furnishes a method for constructing a plane parallel to a given line, AB. In such a construction always construct first a line, CD, parallel to the given line AB, and then pass a plane through CD. Since an infinite number of planes can be passed through CD, it is obvious that this problem is indeterminate unless the required plane must also satisfy other conditions, e.g. to be parallel to a second line or to pass through a given point, or to pass through another line, etc. 487. COR. 1. Through a given straight line AB a plane can be passed parallel to any other given straight line CD; and if the lines are not parallel, only one such plane can be drawn. 488. COR. 2. Through a given point P a plane can be passed parallel to any two given straight lines in space; and B |