EXAMPLE Find a point X, a inches from a given point A and 6 inches from a given line l. The solution is readily obtained by applying the Theorems I and II on page 172. The points of intersection of the line 1, and the circle in the figure evidently satisfy both conditions. Discussion. What is the result (1) if l touches the circle? (2) if the circle cuts 1 and 1? (3) if A is on 1? a 166 A 2 In what way do the relative values of a and b affect the result? EXERCISES 1. Construct a point 2 in. from a given point and 3 in. from a given line. When is the solution impossible? 2. Find a point P which lies in a given line and is equidistant from two given points. Discuss the problem. 3. Construct a point equidistant from two intersecting lines and also at a given distance from one of the lines. 4. Construct a point equidistant from the sides of an angle and at a given distance from the vertex. 5. A tree is to be planted 10 ft. from the front wall and 15 ft. from a corner of a rectangular house. How many solutions are possible? 6. Find in a given circle a point which is equidistant from two given points. 7. Construct a point equidistant from two given concentric circles and also equidistant from two given lines. 8. Find the locus of the center of a circle which (a) passes through two given points; (b) touches each of two given parallels; (c) touches a given line AB at a given point P. 9. Construct a circle which has its center in a given line and passes through two given points. 10. Construct a circle which passes through two given points and has its center equidistant from two given intersecting lines. 11. Construct a point which is at a given distance from a given circle and at a given distance from a secant of that circle. 12. Construct a point which is equidistant from two given parallel lines and from a transversal to those lines. 13. Construct a circle which is tangent to a given line at a given point and has its center on a given line. 14. In the diagram let ABCDrepresent a baseball diamond (square); R, a street; H, a house; E, a corner of the house; Sand T, two trees; F, a fence; P, a circular pond. How would you locate a person who is H P R B E S. F A C T 0 D (a) equidistant from S and T and 5 yd. from CD? (b) 10 yd. from R and 3 yd. from E? (c) 3 ft. from P and 30 yd. from O? (d) equidistant from Rand F? (e) equidistant from AB and CD, and from S and T? (f) 20 yd. from AB and 5 ft. from F? MISCELLANEOUS EXERCISES ON LOCI 1. What is the locus of the vertex of a triangle that has a given base and a given altitude? 2. What is the locus of the vertex of the right angle of a right triangle which has a given hypotenuse? 3. Find the locus of the vertex of a triangle which has a given base and a given vertex angle. 4. What is the locus of the center of a circle which is inscribed in a triangle with a given base and a given vertex angle? 5. What is the locus of the mid-point of a line which is drawn from a given point to a given line? 6. What is the locus of the mid-point of a chord of given length in a circle of given radius? 7. What is the locus of the mid-points of all chords drawn from a given point in a given circle? 8. What is the locus of the point of intersection of the diagonals of a rhombus constructed on a given line as a base? 9. What is the locus of a point equidistant from two given equal circles? 10. A square (equilateral triangle, regular hexagon) is moved along a straight line by revolving it successively about its vertices. What is the locus of one vertex of the square (triangle, hexagon) from one point of contact with the line to its next successive point of contact? C 11. Two stations, A and B, on the shore of a lake are 900 yds. apart. A ship C is observed simultaneously from both stations, A B and the angles CAB and CBA are measured by the observers at each station. In order to determine the path of the ship, additional measurements are made from time to time and are telephoned from one station to the other. Determine the path of the ship if the measurements are as follows: 12. How may the accuracy of a drawing of a circle be tested with a carpenter's square? 13. What locus problem is suggested by a "saw-toothed" roof? 14. Three fortified islands, A, B, C, are so far from land that their guns do not carry to shore. A hostile cruiser is reconnoitering in the vicinity of the islands and the captain wishes to sail around them by the shortest possible route, but always out of range of the guns. Construct the course of the cruiser, using arbitrary values for the ranges of the guns on the islands. 15. A ship, S, approaching land is in the vicinity of three prominent landmarks, A, B, C. The captain wishes to determine how far he is from the shore. He finds that the distance AB subtends at S an angle of 60°, while BC subtends A S C an angle of 95°. From the map the captain finds that AB = 16 mi., and BC = 12 mi. Explain how the length of SB can be found by construction and measurement. (The "three-point problem.") COÖRDINATES. SQUARED PAPER 297. Draw AB to CD. From the intersection point O lay off equal segments on the four rays OA, OB, OC, OD. Through the points of division draw perpendiculars to the lines. A network of squares will be formed. Paper showing such an arrangement of squares is called squared paper. "Inch paper" is ruled into square inches and tenths of an inch; "millimeter paper is ruled into square millimeters and square centimeters. دو B along a line parallel to OC. This process of locating a point by means of two numbers is called plotting the point. The point O is called the origin. The lines AB and CD are called axes. The two numbers are called the coördinates of the point. The distance from O on AB is called the abscissa, and the distance from the line AB is called the ordinate of the point. Distances measured to the right from O on OB are considered as positive (+) numbers, and distances to the left from O on OA are considered as negative (-) numbers. Distances vertically upward from A B are considered as positive, while distances vertically downward from AB are considered negative. Hence the coördinates of P, are (-15,10); those of Pare (-15, - 10); those of Pare (15, -10). 2 EXERCISES 1. Locate on a piece of squared paper four points, and name their coördinates with reference to two convenient axes. 2. Locate the points (5, 6), (— 4, 7), (— 8, — 6), (7, — 5). 3. Plot the following points and connect them so as to form a pattern: (0, 0), (0, 2), (0, 5), (0, 7); (2, 0), (2, 2), (2, 5), (2, 7); (5, 0), (5, 2), (5, 5), (5, 7); (7, 0), (7, 2), (7, 5), (7, 7). 4. Plot the points (4, 5), (— 4, — 5), (0,8). What kind of geometrical figure is determined by these points? 5. What figure is determined by the points (10,7), (6, — 3), (-6, -3), (-2, 7)? 6. A rectangle drawn on squared paper is symmetrical with respect to both axes. One vertex is (12,5). What are the other vertices? 7. Determine, by drawing, the coördinates of the points which are 5 units from the horizontal axis, and at a distance from the origin equal to 13 units. 8. Ascertain, by drawing, the coördinates of the points which are equidistant from the two axes, and at a distance from the origin equal to 10 units. 298. The method of coördinates may be conveniently used to determine the locus of a point when the prescribed condition involves the distances of the point from two perpendicular lines. EXAMPLES 1. The distances x and y of a point from two given perpendicular lines satisfy the equation Determine the locus of the point. Solution. We interpret x and y as coördinates with respect to the given lines as axes. To locate a number of points which satisfy the given condition (1), assume values for a and compute from (1) the corresponding values of y. For example, taking x = 2, then y = 2 × 2 +1 = 5. Hence the point (2, 5) is on the locus. In this way we compute the coördinates of any |