43. Draw a line parallel to the base of a triangle, which shall equal the sum of the lower segments of the sides. Bisect the angles at the base. 44. In any triangle, given one angle, a side adjacent to the given angle, and the difference of the other two sides, to construct the triangle. First, construct a triangle of the given side and difference, with the given angle included. 45. The vertical angle of an oblique-angled triangle inscribed in a circle, is greater or less than a right angle, by the angle contained between the base and the diameter drawn from the extremity of the base. 46. Draw semicircles upon the three sides of a rightangled triangle, and the area of the crescents thus formed will equal that of the triangle. 47. Draw a diameter through P, a point within a circle, and the two segments of the diameter will measure the least and the greatest distance from P to the circumference. 48. If from the middle point P, of an arc NS, the chords PD and PC are drawn cutting the chord NS in A and B, the opposite angles of ABCD are supplementary. 49. Through the point P, without a given angle, draw a line cutting the sides of the angle, so that the triangle formed shall have a given perimeter. 50. From the centers of two opposite sides of any quadrilateral, draw lines to the centers of the diagonals, and the figure thus formed is a parallelogram. 51. Upon AC, the diagonal of a square ABCD, mark off AP equal to AB, and draw PO perpendicular to the diagonal and terminating in the side, and prove BOOP PC (Sec. X, Ex. 17). = 52. Two circles touch each other at T, and have a common tangent AB. The chords AT and BT form a right angle. Connect the centers, and draw radii to A and B (Theorems XXXI and XXXV, Book I). 53. The square described upon a line is equal to the squares upon any two segments of the line increased by twice the rectangle contained by those segments. 54. Two chords of a circle intersect at right angles. Prove that the squares of the four segments are together equal to the square of the diameter. (Theo. XIII, Cor. 1, Book II). Sch. Upon review, pupils should at times be incited to disregard the methods suggested in the lists of Exercises, and search for others. Read the preface carefully before beginning the review. SOLID GEOMETRY. BOOK III. SECTION XIV.-PLANES AND LINES. DEFINITIONS. 1. In solid geometry the magnitudes treated of are not limited to one plane. 2. A plane has already been defined to be a surface such that, if any two of its points be connected by a straight line, the line will lie wholly within the surface. 3. A straight line is perpendicular to a plane when it is perpendicular to every straight line of that plane which it can meet. The plane is also perpendicular to the line. 4. The line in which two planes cut one another is called their intersection. Cor. 1.-The intersection of two planes is a straight line; for, if any two points in the intersection be taken, the straight line joining these points must be in both planes (?). Cor. 2.-Two planes have no common point without their intersection (?). 5. The angle of inclination of two planes is that contained by any two straight lines perpendicular to their intersection, one in each plane. If the line in each plane is perpendicular to the other plane, the two planes are said to be perpendicular to each other, the angle of inclination being a right angle. 6. The diedral is the angle formed by two planes which cut one another. By simply opening a book at various angles, we have examples of acute, right, and obtuse diedrals. 7. Two planes are said to be parallel when their intersections by every third plane which can cut them are parallel. Cor. 1.-Two parallel planes can never meet; for, if they were to meet in any point, their intersections by a third plane passing through that point would also meet. By the definition, all such intersections are parallel. Cor. 2.-Two planes which are parallel to a third are parallel to each other. Cor. 3.-Two planes which are perpendicular to the same straight line are parallel to each other. 8. The projection of a point upon a plane is the foot of a perpendicular let fall from the point to the plane. The projection of a line contains the foot of every perpendicular which can be drawn from the line to the plane; that is, it is the locus of the projections of all its points. Cor. The projection of a straight line upon a plane is a straight line. Ques. What exception to this statement? 9. The inclination of a line to a plane is the angle it makes with its projection on the plane. 10. The position of a plane is determined by its containing three points not in a straight line, or by its containing a straight line and a point without it. The plane may be turned upon the line as an axis until it embraces the given point. 11. The axis of a circle is a straight line perpendicular to the circle at its center. Cor. Any point in the axis is equally distant from all points in the circumference (?). Ques. What is the locus of all the points equally distant from three given points? THEOREM I. Two straight lines perpendicular to the same plane are parallel. Let AB and CD be two perpendiculars to the plane EF. It is to be shown that they are parallel. Join BD. Now, AB and CD being perpendicular to the plane, are also perpendicular to the line BD (Def. 3). C Also, a plane perpendicular to EF, and passing through BD, will contain both AB and CD (Def. 5). But straight lines in the same plane, and perpendicular to the same straight line, are parallel (Cor. 2, Theo. IV, Book I.) Therefore, AB is parallel to CD. That is, two straight lines, etc. Cor. If one of two parallels is perpendicular to a plane, the other is also perpendicular to the plane. Ques. Can two lines cut a plane at the same angle and not be parallel? E. G.-11. |