6. Through any point on the bisector of any angle AOB draw a perpendicular to the bisector, cutting OA in C and OB in D. Prove that DC is bisected by OP, that OC is equal to OD, and that angle OCD is equal to angle ODC. 7. On the sides of an angle AOB take points C and D equally distant from O. Draw CD. Prove that angle OCD is equal to angle ODC. 8. Show that the lines drawn from the extremities of a line segment to any point on the perpendicular bisector of the segment make equal angles with the perpendicular and with the segment. 9. In the following figures are shown the geometric foundations for various designs used in window tracery. Many variations may be obtained by coloring certain portions of the drawings, and by other artistic devices. Let the student draw the designs, adding or inventing embellishments if desired. PARALLEL LINES 76. Parallel lines. Two (unlimited) straight lines in the same plane which do not meet are said to be parallel (||). 77. Two straight lines in the same plane which are perpendicular to the same straight line are parallel. But no more For, if they are not parallel they will meet. than one perpendicular can be drawn through a given external point to a given straight line (§ 53). 78. FUNDAMENTAL PROPOSITION. Through a given point not on a given straight line one straight line, and only one, can be drawn parallel to the given line. 79. PROBLEM. Through a given point P not on a given straight line AB, to draw a line parallel to that line. (Let the student make the construction by the use of compasses and straightedge.) 80. For drawing parallel lines practical draftsmen use a T-square in connection with a drafting board (see Fig. 1, p. 1) or a draftsman's triangle (see Fig. 29). 81. Two straight lines which are parallel to the same straight line are parallel to each other. FIG. 29. For, if they are not parallel, they will meet; but two intersecting lines cannot both be parallel to the same straight line (§ 78). 82. A straight line which is perpendicular to one of two parallel lines is perpendicular to the other also. For, let AB and CD be two parallel lines, and let EF be perpendicular to AB, cutting AB and CD in G and H, respectively. A perpendicular to EF at H is parallel to AB, and coincides with CD. Why? EXERCISE Under what conditions is Ex. 1, § 59, impossible? : 83. Two straight lines in the same plane which are perpendicular respectively to two intersecting lines must meet. The lines cannot be parallel, for then each would be perpendicular to both of the given intersecting lines (§ 53). 84. Two parallel lines are everywhere equally distant. For, two parallel lines are symmetric with respect to any perpendicular (§ 50). 1. Two straight streets each 60 feet wide intersect at an angle of 45°. The building lines are 20 feet from the curbing. Draw a plan of the corner. 2. Two straight streets each 60 feet wide intersect at an angle of 135°. Draw a plan showing how to locate the corner stone of a building 20 feet from one street and 30 feet from the other. 3. A straight railway passes two miles from a town. A place is four miles from the town and one mile from the railway. Find by construction the places that answer this description. 4. A watch dog is tethered to a stretched wire 30 feet long by a rope. The rope is 15 feet long and is fastened to the wire by a ring free to move along the whole length of the wire. Draw a plan showing the dog's complete range, assuming that the rope when stretched is horizontal. 5. A horse is tethered to one corner of a house 30 feet wide by 60 feet long, by a rope 100 feet long. Draw a plan showing the horse's complete range. 6. A and B are two points equally distant from and on the same side of a straight line CD. Show that the straight line determined by A, B is parallel to CD. 7. The following figures are examples of pattern designs for mosaics, tiles, floorcloths, etc. The constructions are based on "the square net," that is, a series of parallel lines equal distances apart, crossed by a like series of parallels at right angles to the first. MORE ABOUT THE CIRCLE. TANGENTS 86. In § 44 it was proved that a circle is symmetric with respect to any diameter. As an immediate consequence of this proposition we infer the following: I. The perpendicular drawn to a chord through the center of a circle bisects the chord and the arcs subtended (cut off) by the chord. 87. II. The straight line drawn through the center of a circle and bisecting a chord (which is not a diameter) is perpendicular to the chord. D A OM B FIG. 31. 88. III. The perpendicular drawn to a chord through its midpoint passes through the center of the circle. 89. PROBLEM. To describe a circle through three given points A, B, and C not on a straight line. C A B FIG. 32. Construction. If A, B, C are on a circle, the segments AB and BC are chords. Reference to the preceding paragraph shows that the problem will be solved by drawing the perpendicular These lines meet in a point which is bisectors of AB and BC. equally distant from A, B, and C. Let the pupil explain. See §§ 83, 58. |