11. Prove that if a line is parallel to one plane and perpendicular to another, the two planes are perpendicular to 13. Describe how to locate a point that is 4 in. from face of a triedral angle, 5 in. from another face, and 3 in. from the third face. 14. If a series of parallel planes cut all the edges of a triedral angle, the intersections of the planes with the faces form similar triangles. 15. If from P, any point within the diedral angle A-BC-D, PM and PN are drawn perpendicular to the faces BD and AC respectively, and MS is drawn perpendicular to AC, then NS is perpendicular to BC. 16. The three planes bisecting the diedral angles of a triedral angle meet in a line. B SUGGESTION. Show that two of them meet in a line, and then show that this line lies in the third plane. Compare this exercise with Exercise 13, page 308. 17. All points within a triedral angle and equally distant from its three faces, lie in the line of intersection of the planes that bisect the diedral angles. A N C D E 18. The planes through the bisectors of the face angle of a triedral angle and perpendicular to the planes of the respective faces, meet in a line. CHAPTER VII POLYEDRONS, PRISMS, CYLINDERS 613. In the present and following chapters will be considered some of the solids most commonly observed in nature, and very frequently used in architecture, engineering, and the arts. Besides finding the areas and volumes of these solids it will be necessary to investigate the relations of their parts, and to study the plane figures formed when the solids are cut by planes. 614. Polyedrons. A polyedron is a solid entirely bounded by planes. The intersections of the bounding planes are the edges of the polyedron. The points in which the edges intersect are the vertices. The polygons bounded by the edges are the faces. The faces taken together make up the surface, and the area of this surface is the area of the polyedron. The amount of space enclosed by the surface is the volume of the polyedron. 615. Sections. The plane figure formed on a plane passing through a solid, and bounded by the intersections of the plane with the surface of the solid is called a section of the solid. It is evident that the sec tion of a solid is a closed figure (§ 146), and that the section of a polyedron is a polygon. 616. Convex Polyedrons. A polyedron is convex if every section of it is a convex polygon. Otherwise it is concave. Only convex polyedrons are considered unless otherwise stated. 617. Prismatic and Cylindrical Surfaces. A moving straight line that always remains parallel to its original position, and intersects a fixed straight line, generates a plane. Why? A moving straight line that always remains parallel to its original position, and intersects a broken line not coplanar with it, generates a prismatic surface. A moving straight line that always remains parallel to its original position, and intersects a plane curved line not coplanar with it, generates a cylindrical surface. AB is the original position of the moving line, and MN is the fixed line. 618. The moving line is called the generatrix, and the fixed line or curve the directrix. The generatrix in any position is called an element of the surface generated. 619. Closed Surface. If the directrix is a closed line, the prismatic or cylindrical surface is closed. ΕΙ 620. Sections of Prismatic and Cylindrical Surfaces. plane cutting all the elements of a closed prismatic or cylin drical surface cuts the surface in a closed line. The figure bounded by this closed line is a section. If the cutting plane is perpendicular to an element, the section is a right section; if not, it is an oblique section. Two sections made by parallel planes are parallel sections. In the figure, ABCDE is a right section. A A PRISMS AND CYLINDERS 621. A prism is the solid formed by a closed prismatic surface and two parallel cutting planes. 622. A cylinder is the solid formed by a closed cylindrical surface and two parallel cutting planes. 623. The sections formed by the cutting planes are called the bases of the prism or cylinder. It follows from § 620 that the bases of a prism are polygons. 624. The polygons formed on the pris matic surface between the bases are called the lateral faces. Intersections of the lateral faces are lateral edges, and intersections of the lateral faces with the bases are base edges. 625. The altitude of a prism, or cylinder, is the perpendicular distance between its bases. 626. The lateral area of a prism, or cylinder, is the prismatic, or cylindrical, surface between the bases. The total area is the lateral area together with the areas of the bases. In the figure read the bases, faces, lateral edges, base edges, altitude. F A IN G H IM B C 627. The following facts about prisms and cylinders are readily proved by the definitions and previous theorems: (1) The lateral edges of a prism, or the elements of a cylinder, are equal. (2) The lateral faces of a prism are parallelograms. (3) The right section of a prism, or cylinder, is perpendicular to all the lateral edges, or elements. (4) The section of a prism made by a plane parallel to a lateral edge is a parallelogram. (5) The section of a cylinder made by a plane containing two elements is a parallelogram, 628. Theorem. Parallel sections of a prism are congruent. Given the prism MN, and the parallel sections AD and A'D'. To prove polygon AD polygon A'D'. N Why? A B C Also EAB= [ E'A'B', [ ABC= [ A'B'C', etc. § 567 M The two polygons AD and A'D' are then mutually equilateral and mutually equiangular. They then can be proved congruent by superposition. .. polygon AD polygon A'D'. 629. Theorem. Parallel sections of a cylinder are congruent. Given the cylinder MN, and the parallel sections P and Q made by the planes. To prove PQ. Proof. Take two points A and B on the perimeter of P, and let C be any third point of that perimeter. Draw the elements through A, B, and C to the points A', B', and C' respectively in the perimeter of Q; and draw AB, BC, CA, A'B', B'C', and C'A'. M N Why? And, if P is superposed on Q with AB coinciding with A'B', C will fall upon C'. But C is any point in the perimeter of P. Why? Hence every point in the perimeter of P will fall upon a point of Q, and conversely. .. P=Q. 630. Theorem. Every section of a prism, or cylinder, parallel to the bases is congruent to them. |