LOCI 611. As has been assumed on the previous pages, the idea of a locus here is the same as in plane geometry. Here, too, the proof of a locus theorem must, as in plane geometry, consist of two parts: (1) That all points in the figure satisfy the given conditions. (2) That all points that satisfy the given conditions are in the figure. In the place of (2) one may prove that all points not in the figure do not satisfy the given conditions. 612. Jn plane geometry it was noticed that usually a locus satisfying one condition is a line or group of lines, and a locus satisfying two conditions is a point or group of points. There, however, a further condition is assumed, namely, that the locus is confined to a plane. In solid geometry usually, one condition will confine the locus to a surface or group of surfaces, two conditions will confine the locus to a line or group of lines, and three conditions will confine the locus to a point or group of points. Here, as in plane geometry, one of the greatest benefits derived from the study of loci is through the imaging and constructing figures, rather than through the proofs of the loci theorems. This, however, does not mean that the proofs should be neglected. EXERCISES 1. What is the locus of points equally distant from two given points? 2. What is the locus of points equally distant from three given points? 3. What is the locus of points equally distant from four given points? 4. What is the locus of points equally distant from two intersecting lines and in a given plane? 5. What is the locus of the end point of a line-segment of fixed length, that moves so as to remain parallel to a given line and have one end in a given plane? 6. What is the locus of points in a given plane equally distant from three given points not in a line? 7. What is the locus of points equally distant from two given parallel lines? 8. What is the locus of the middle point of a line-segment of fixed length, that moves so as to have its end points in two parallel planes? 9. What is the locus of a point equally distant from two parallel planes and equally distant from the faces of a diedral angle? 10. What is the locus of all lines that pass through a given point and are parallel to a given plane not containing the point? 11. What is the locus of the points within a diedral angle and equally distant from its two faces? 12. What is the locus of all points that are twice as far from a line in a plane as from the plane? 13. What is the locus of the points within a triedral angle and equally distant from its three faces? 14. What is the locus of the points equally distant from the edges of a triedral angle? 15. What is the locus of a line making equal angles with each of two intersecting lines? What other exercise in this list has the same locus as this? 16. What is the locus of points in one of two given non-coplanar lines equally distant from two given points in the other? 17. What is the locus of a point that moves so that its distance from one of two given parallel planes is to its distance from the other as 1 : 3? 18. What is the locus of a point whose distances from the faces of a diedral angle are respectively 5 in. and 8 in.? 19. What is the locus of the projections of a given point upon the planes containing a given line? Many questions concerning loci cannot be discussed in elementary geometry as they lead to forms not there considered, for in elementary geometry only a limited number of forms such as lines, circles, planes. cones, cylinders, and spheres are considered. The question: What is the locus of the middle points of all transversals of two non-coplanar lines? leads to none of these. In general, the locus of points equally distant from two different elements, as a point and a line, a line and a plane, or two non-coplanar lines cannot be considered in elementary geometry. QUESTIONS 1. State the different relations that two lines may have to each other. That a line may have to a plane. That two planes may have to each other. 2. State the different ways of determining that a straight line lies in a plane. 3. What is the intersection of a line and a plane? Of two planes? Of three planes? 4. State the different ways of determining that a line is parallel to a plane. 5. State the different ways of determining that a line is perpendicular to a plane. 6. State the different ways of determining that two planes are parallel. 7. State the different ways of determining that two planes are perpendicular. 8. How many intersecting lines can be parallel to the same line? To the same plane? 9. How many intersecting planes can be parallel to the same line? To the same plane? 10. Must planes be parallel if they contain parallel lines? Are lines necessarily parallel if they are in parallel planes? 11. How many lines through a point can be perpendicular to the same line? To the same plane? 12. How many intersecting planes can be perpendicular to the same line? To the same plane? 13. What are the relations of the following to each other: (1) lines parallel to the same line; (2) planes parallel to the same plane; (3) lines parallel to the same plane; (4) planes parallel to the same line; (5) lines perpendicular to the same line; (6) lines perpendicular to the same plane; (7) planes perpendicular to the same line; (8) planes perpendicular to the same plane? 14. If two planes are perpendicular to each other, what lines in one are perpendicular to the other? 15. Can the projection of a straight line upon a plane be a curve? Can the projection of a circle be a straight line? Explain. 16. Can the projection of a square be a square? A rectangle? A parallelogram? Explain. GENERAL EXERCISES COMPUTATION 1. Two fixed points, A and B, are 10 in. apart. A point P moves so as to keep 8 in. from A and B. Find the length of the locus of P. 2. Two fixed points, A and B, are 12 in. apart. A point P moves so as to keep 10 in. from A and 4 in. from B. Find the radius of the circle generated. Ans. 3.13+ in. 3. A 20-foot pole is placed obliquely in a river. One end of the pole is on the bottom of the river and the other end is 3 ft. above the surface. Find the depth of the river if the length of the part of the pole under water is 15 ft. 4. Plane Q is perpendicular to plane P. shortest line AED that can be drawn from a Q to a point D in P if AB=5 ft., CD= 10 ft., and BC = 20 ft. SUGGESTION. Find the point A in Consider the figure when Q is turned into the same plane as P. 5. The room shown in the figure has a length of 40 ft., a width of 20 ft., and a height of 12 ft. Find the length of the shortest line in the walls of the room from B to H. 6. The sum of the face angles of a triedral angle is 148°. What is the greatest value a face angle can have? 7. In the figure of a hammock support with the dimensions as given, find the length A of AB if AB=CB=DE=FE. B 8. The roof of a house makes an angle of 30° with the plane of the plates. The roof is 18 ft. by 30 ft. Find the area of its projection upon the plane of the plates. 9. In the figure of a cube an edge is 18 in. and BE is 10 in. Find the area of ABCD. 10. Having given the dimensions as shown in the figure, find the area of section S to two decimal places. All the face planes that meet are perpendicular to each other, 30' F 14 B 20 1 1. If a series of parallel planes intercept equal segments on one straight line, they intercept equal segments on any other straight line that they intersect. 2. If a straight line and a plane are each perpendicular to the same straight line, they are parallel. 3. If two planes are perpendicular to each other, a line perpendicular to one and not in the other is parallel to the other. 4. Prove that the sides of an isosceles triangle make equal angles with any plane in which the base lies. 5. From a point within a triedral angle perpendiculars are drawn to the three faces. These perpendiculars are the edges of a second triedral angle. Prove that the face angles of one are supplementary to the diedral angles of the other. 6. Three planes, M, N, and Q, each perpendicular to the other two pass through a common point. Show that a point 5 in. from M, 8 in. from N, and 10 in. from Q may be in any one of eight positions. 7. Given two non-coplanar lines, to construct a plane upon which the projections of the two lines will be parallel. 8. A ray of light from the source L is reflected L by the mirror M and enters the eye at E. The path travelled by the light is the shortest possible path from L to the mirror and then to E. Show how to determine where the ray of light strikes the mirror. Α B E M 9. Three non-coplanar lines meet at a point. Construct a line that will make equal angles with the three lines. 10. Given a polyedral angle with four faces, to construct a plane that will intersect the four faces so that the intersection will be a parallelogram. SUGGESTION. Pass a plane parallel to one face and intersecting the opposite face. Determine a line in the first face equal to this intersection and parallel to it. These two lines are parallel and hence determine a parallelogram. |