PROBLEMS. The most useful general precept that can be given, to aid the student in his search for the solution of a problem, is the following: Suppose the problem solved, and construct a figure accordingly; study the properties of this figure, drawing auxiliary lines when necessary, and endeavor to discover the dependence of the problem upon previously solved problems. This is an analysis of the problem. The reverse process, or synthesis, then furnishes a construction of the problem. In the analysis, the student's ingenuity will be exercised especially in drawing useful auxiliary lines; in the synthesis, he will often find room for invention in combining in the most simple form the several steps suggested by the analysis. The analysis frequently leads to the solution of a problem by the intersection of loci. The solution may turn upon the determination of the position of a particular point. By one condition of the problem it may appear that this required point is necessarily one of the points of a certain line; this line is a locus of the point satisfying that condition. A second condition of the problem may furnish a second locus of the point; and the point is then fully determined, being the intersection of the two loci. Some of the following problems are accompanied by an analysis to illustrate the process. 28. To determine a point whose distances from two given intersecting straight lines, AB, A'B', are given. C A P P2 ·E' points at a given distance from A'B' consists of two parallels, C'E' and D'F', each at the given distance from A'B'. The required point must be in both loci, and therefore in their intersection. There are in this case four intersections of the loci, and the problem has four solutions. Construction. At any point of AB, as A, erect a perpendicular CD, and make AC= AD = the given distance from AB; through C and D draw parallels to AB. In the same manner, draw parallels to A'B' at the given distance A'C' A'D'. The intersec tion of the four parallels determines the four points P1, P2, P3, P41 each of which satisfies the conditions. 29. Given two perpendiculars, AB and CD, intersecting in O, to construct a square, one of whose angles shall coincide with one of the right angles at O, and the vertex of the opposite angle of the square shall lie on a given straight line EF. (Two solutions.) 30. In a given straight line, to find a point equally distant from two given points without the line. 31. To construct a square, given its diagonal. 32. Through a given point P within a given angle, to draw a straight line, terminated by the sides of the angle, which shall be bisected at P. (v. Exercise 28, Book I.) 33. Given two straight lines which cannot be produced to their intersection, to draw a third which would pass through their intersection and bisect their contained angle. Suggestion. Find two points equidistant from the two lines. (v. I., Proposition XIX.) 34. Given the middle point of a chord in a given circle, to draw the chord. 35. To draw a tangent to a given circle which shall be parallel to a given straight line. 36. To draw a tangent to a given circle, such that its segment intercepted between the point of contact and a given straight line shall have a given length. Suggestion. The tangent, the radius drawn to the point of contact, and a line drawn from the centre to the end of the tangent form a right triangle, two of whose sides are known. A simple construction gives the hypotenuse. In general there are four solutions. Show when there will be but two; also, when no solution is possible. 37. Through a given point within or without a given circle, to draw a straight line, intersecting the circumference, so that the intercepted chord shall have a given length. (Two solutions.) (v. Exercise 23 and Section 78.) 38. Construct an angle of 60°, one of 120°, one of 30°, one of 150°, one of 45°, and one of 135°. 39. Construct a triangle, given the base, the angle opposite to the base, and the altitude. M A N B с Analysis. Suppose BAC to be the required triangle. The side BC being fixed in position and magnitude, the vertex A is to be determined. One locus of A is an arc of a segment, described upon AB, containing the given angle. Another locus of A is a straight line MN drawn parallel to BC, at a distance from it equal to the given altitude. Hence the position of A will be found by the intersection of these two loci, both of which are readily constructed. Limitation. If the given altitude were greater than the perpendicular distance from the middle of BC to the arc BAC, the arc would not intersect the line MN, and there would be no solution possible. The limits of the data within which the solution of any problem is possible should always be determined. 40. Construct a triangle, given the base, the medial line to the base, and the angle opposite to the base. 41. With a given radius, describe a circumference, 1st, tangent to two given straight lines; 2d, tangent to a given straight line and to a given circumference; 3d, tangent to two given circumferences; 4th, passing through a given point and tangent to a given straight line; 5th, passing through a given point and tangent to a given circumference; 6th, having its centre on a given straight line, or a given circumference, and tangent to a given straight line, or to a given circumference. (Exercises 19, 20, 21.) 42. Describe a circumference, 1st, tangent to two given straight lines, and touching one of them at a given point (Exercises 17, 18); 2d, tangent to a given circumference at a given point and tangent to a given straight line; 3d, tangent to a given straight line at a given point and tangent to a given circumference (Exercise 18); 4th, passing through a given point and tangent to a given straight line at a given point; 5th, passing through a given point and tangent to a given circumference at a given point. 43. Draw a straight line equally distant from three given points. When will there be but three solutions, and when an indefinite number of solutions? 44. Inscribe a straight line of given length between two given circumferences, and parallel to a given straight line. (v. Exercise 25.) BOOK III. PROPORTIONAL LINES. SIMILAR FIGURES. THEORY OF PROPORTION. 1. DEFINITION. One quantity is said to be proportional to another when the ratio of any two values, A and B, of the first, is equal to the ratio of the two corresponding values A' and B', of the second; so that the four values form the proportion (II., 36) This definition presupposes two quantities, each of which can have various values, so related to each other that each value of one corresponds to a value of the other. An example occurs in the case of an angle at the centre of a circle and its intercepted arc. The angle may vary, and with it also the arc; but to each value of the angle there corresponds a certain value of the arc. It has been proved (II., Proposition XII.) that the ratio of any two values of the angle is equal to the ratio of the two corresponding values of the arc; and, in accordance with the definition just given, this proposition would be briefly expressed as follows: "The angle at the centre of a circle is proportional to its intercepted arc." 2. Definition. One quantity is said to be reciprocally proportional to another when the ratio of two values, A and B, of the first, is equal to the reciprocal of the ratio of the two corresponding values, A' and B', of the second, so that the four values form the proportion For example, if the product p of two numbers, x and y, is given, so that we have then x and y may each have an indefinite number of values, but as x increases y diminishes. If, now, A and B are two values of x, while A' and B' are the two corresponding values. of y, we must have АХА B × B' = p, whence, by dividing one of these equations by the other, that is, two numbers whose product is constant are reciprocally proportional. 3. Let the quantities in each of the couplets of the proportion be measured by a unit of their own kind, and thus expressed |
