GEOMETRICAL EXERCISES ON BOOK XI. THEOREM I. Define the projection of a straight line on a plane; and prove that if a straight line be perpendicular to a plane, its projection on any other plane, produced if necessary, will cut the common intersection of the two planes at right angles. Let AB be any plane and CEF another plane intersecting the former at any angle in the line EF; and let the line GH be perpendicular to the plane CEF. Draw GK, HL perpendicular on the plane AB, and join LK, then EF, the intersection of the two planes, is perpendicular to every plane passing through GH, and therefore the projecting plane GHKL is perpendicular to the plane CEF; but the projecting plane GHLK is perpendicular to the plane AB; (constr.) hence the planes CEF, and AB are each perpendicular to the third plane GHLK; therefore EF, the intersection of the planes AB, CEF, is perpendicular to that plane; and consequently, EF is perpendicular to every straight line which meets it in that plane; but EF meets LK in that plane. Wherefore, EF is perpendicular to KL, the projection of GH on the plane AB. THEOREM II. Prove that four times the square described upon the diagonal of a rectangular parallelopiped, is equal to the sum of the squares described on the diagonals of the parallelograms containing the parallelopiped. Let AD be any rectangular parallelopiped; and AD, BG two diagonals intersecting one another; also AG, BD, the diagonals of the two opposite faces HF, CE. Then it may be shewn that the diagonals AD, BG, are equal; as also the diagonals which join CF and HE: and that the four diagonals of the parallelopiped are equal to one another. The diagonals AG, BD of the two opposite faces HF, CE are equal to one another: also the diagonals of the remaining pairs of the opposite faces are respectively equal. And since AB is perpendicular to the plane CE, it is perpendicular to every straight line which meets it in that plane, And the therefore AB is perpendicular to BD, and consequently ABD is a right-angled triangle. Similarly, GDB is a right-angled triangle. square of AD is equal to the squares of AB, BD, (1. 47.) also the square of BD is equal to the squares of BC, CD, therefore the square of AD is equal to the squares of AB, BC, CD; similarly the square of BG or of AD is equal to the squares of AB, BC, CD. Wherefore the squares of AD and BG, or twice the square of AD, is equal to the squares of AB, BC, CD, AB, BC, CD; but the squares of BC, CD are equal to the square of BD, the diagonal of the face CE; similarly, the squares of AB, BC are equal to the square of the diagonal of the face HB; also the squares of AB, CD, are equal to the square of the diagonal of the face BF; for CD is equal to BE. Hence, double the square of AD is equal to the sum of the squares of the diagonals of the three faces HF, HB, BC. In a similar manner, it may be shewn, that double the square of the diagonal is equal to the sums of the squares of the diagonals of the three faces opposite to HF, HB, BC. Wherefore, four times the square of the diagonal of the parallelopiped is equal to the sum of the squares of the diagonals of the six faces. PROBLEMS. 1. Two of the three plane angles which form a solid angle, and also the inclination of their planes being given, to find the third plane angle. 2. Having three points given in a plane, find a point above the plane equidistant from them. 3. To describe a circle which shall touch two given planes, and pass through a given point. 4. Two triangles have a common base, and their vertices are in a straight line perpendicular to the plane of the one; there are given the vertical angle of the other, the angles made by each of its sides with the plane of the first and the distance of the vertices of the two triangles, to find the common base. 5. Find the distance of a given point from a given line in space. 6. Find a point in a given straight line such that the sums of its distances from two given points (not in the same plane with the given straight line) may be the least possible. 7. Draw a line perpendicular to two lines which are not in the same plane. 8. Two planes being given perpendicular to each other draw a third perpendicular to both. 9. Required the perpendicular from the vertex upon the base of a triangular pyramid, all the sides of which are equilateral triangles of a given area. 10. Given the lengths and positions of two straight lines which do not meet when produced and are not parallel; form a parallelopiped of which these two lines shall be two of the edges. 11. How many triangular pyramids may be formed whose edges are six given straight lines, of which the sum of any three will form a triangle? 12. Bisect a triangular pyramid by a plane passing through one of its angles, and cutting one of its sides in a given direction. 13. Define a cube. When this solid is cut by a plane obliquely to any of its sides, the section will be an oblong, always greater than the side, if made by cutting opposite sides. Draw a plane cutting two adjacent sides, so that the section shall be equal and similar to the side. THEOREMS. 3. If two straight lines are parallel, the common section of any two planes passing through them is parallel to either. 4. If three limited straight lines be parallel, and planes pass through each two of them, and the extremities be joined, a prism will be formed, the ends of which will be parallel if the straight lines be equal. 5. If two straight lines be parallel, and one of them inclined at any angle to a plane, the other also shall be inclined at the same angle to the same plane. 6. Parallel planes are cut by parallel straight lines at the same angle. 7. If from a point A above a plane, straight lines AB, AC be drawn, meeting it in B and C, of which AB is perpendicular to the plane, and AC perpendicular to a straight line DC in that plane, and CB be joined, CB shall be perpendicular to DC. 8. Three straight lines not in the same plane, but parallel to and equidistant from each other, are intersected by a plane, and the points of intersection joined; shew when the triangle thus formed will be equilateral and when isosceles. 9. If two straight lines in space be parallel, their projections on any plane will be parallel. 10. Of all the angles, which a straight line makes with any straight lines drawn in a given plane to meet it, the least is that which measures the inclination of the line to the plane. 11. Three parallel straight lines are cut by parallel planes, and the points of intersection joined, the figures so formed are all similar and equal. 12. If a straight line be at right angles to a plane, the intersection of the perpendiculars let fall from the several points of that line, on another plane, is a straight line which makes right angles with the common section of the two planes. 13. If two straight lines be cut by four parallel planes, the two segments, intercepted by the first and second planes, have the same ratio to each other as the two segments intercepted by the third and fourth planes. 14. If there be two straight lines which are not parallel, but which do not meet, though produced ever so far both ways, shew, that two parallel planes may be determined so as to pass, the one through the one line, the other through the other; and that the perpendicular distance of these planes is the shortest distance of any point that can be taken in the one line from any point taken in the other. 15. If from a point above the plane of a circle straight lines be drawn to the circumference there will be only two of them equal in length, and they will be equidistant from the shortest and longest, and on opposite sides. What is the exception to this proposition? 16. Two planes intersect each other, and from any point in one of them a line is drawn perpendicular to the other, and also another line perpendicular to the line of intersection of both; shew that the plane which passes through these two lines is perpendicular to the line of intersection of the plane. 17. Two points are taken on a wall and joined by a line which passes round a corner of the wall. This line is the shortest when its parts make equal angles with the edge at which the parts of the wall meet. 18. If four straight lines in two parallel planes be drawn, two from one point and two from another, and making equal angles with another plane perpendicular to these two, then if the first and third be parallel, the second and fourth will be likewise. 19. If, round a line which is drawn from a point in the common section of two planes at right angles to one of them, a third plane be made to revolve, shew that the plane angle made by the three planes is then the greatest, when the revolving plane is perpendicular to each of the two fixed planes. 20. The content of a rectangular parallelopipedon whose length is any multiple of the breadth and breadth the same multiple of the depth is the same as that of the cube whose edge is the breadth. 21. A rectangular parallelopiped is bisected by all the planes drawn through the axis of it. 22. If a straight line be divided into two parts, the cube of the whole line is equal to the cubes of the two parts together with thrice the right parallelopiped contained by their rectangle and the whole line. 23. If a right-angled triangular prism be cut by a plane, the volume of the truncated part is equal to a prism of the same base and of height equal to one third of the sum of the three edges. 24. If a four-sided solid be cut off from a given cube, by a plane passing through the three sides which contain any one of its solid angles, the square of the number of standard units in the base of this solid, shall be equal to the sum of the squares of the numbers of similar units, contained in each of its sides. 25. In an oblique parallelopiped the sum of the squares of the four diagonals equals the sum of the squares of the twelve edges. 26. If any point be taken within a given cube, the square described on its distance from the summit of any of the solid angles of the cube, is equal to the sum of the squares described on its several perpendicular distances from the three sides containing that angle. 27. Shew that a cube may be cut by a plane, so that the section shall be a square greater in area than the face of the cube in the proportion of 9 to 8. 28. Why cannot a sheet of paper be made to represent the vertex of a pyramid, without folding? 29. A, B are two fixed points in space, and CD a constant length of a given straight line; prove that the pyramid formed by joining the four points A, B, C, D is always of the same magnitude, on whatever part of the given line CD be measured. 30. Define similar rectilineal figures; and prove, that if a pyramid with a polygon for its base be cut by a plane parallel to the base, the section will be a polygon similar to the base. |