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CHAPTER 11 Locus
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Fig. 11-18
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11.9. Locate each of the following:
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(11.4)
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(a) Treasure that is buried 5 ft from a straight fence and equidistant from two given points where the fence meets the ground (b) Points that are 3 ft from a circle whose radius is 2 ft and are equidistant from two lines which are parallel to each other and tangent to the circle (c) A point equidistant from the three vertices of a given triangle (d) A point equidistant from two given points and equidistant from two given parallels (e) Points equidistant from two given intersecting lines and 5 ft from their intersection (f) A point that is equidistant from the sides of an angle and 1 in from their intersection 2 11.10. Locate the point or points which satisfy the following conditions with respect to ^ABC in Fig. 11-19: (a) Equidistant from its sides (b) Equidistant from its vertices (c) Equidistant from A and B and from AB and BC (d) Equidistant from BC and AC and 5 units from C (e) 5 units from B and 10 units from A (11.4)
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Fig. 11-19
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Fig. 11-20
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11.11. Excluding points lying outside square ABCD in Fig. 11-20, how many points are there that are (a) Equidistant from its vertices (b) Equidistant from its sides (c) 5 units from E and on one of the diagonals (d) 5 units from E and equidistant from AD and BC (e) 5 units from FG and equidistant from AB and CD (f) 20 units from A and 10 units from B 11.12. Prove that the locus of points equidistant from the sides of an angle is the bisector of the angle.
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(11.4)
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Analytic Geometry
12.1 Graphs
A number line is a line on which distances from a point are marked off in equal units, positively in one direction and negatively in the other. The origin is the zero point from which distances are measured. Figure 12-1 shows a horizontal number line.
+5 II +4 +3 +2 +1 5 4 3 2 1 3 Q( 4, 3) 4 1 2 3 4 5
+4 P(+4, +3) +3
Origin
O +1 +2 +3 +4 +5
2 1 0 1 2 Horizontal Number Line
III y
Fig. 12-1
Fig. 12-2
The graph shown in Fig. 12-2 is formed by combining two number lines at right angles to each other so that their zero points coincide. The horizontal number line is called the x-axis, and the vertical number line is the y-axis. The point where the two lines cross each other is, again, called the origin. A point is located on a graph by its coordinates, which are its distances from the axes. The abscissa or x-coordinate of a point is its distance from the y-axis. The ordinate or y-coordinate of a point is its distance from the x-axis. When the coordinates of a point are stated, the x-coordinate precedes the y-coordinate. Thus, the coordinates of point P in Fig. 12-2 are written (4, 3); those for Q are ( 4, 3). Note the parentheses. The quadrants of a graph are the four parts cut off by the axes. These are numbered I, II, III, and IV in a counterclockwise direction, as shown in Fig. 12-2.
SOLVED PROBLEMS
12.1 Locating points on a graph Give the coordinates of the following points in Fig. 12-3:
CHAPTER 12 Analytic Geometry
4 3 P 2 1
R S x
1 2 3 4 4 3 2 1 1 N 2 R G L 4 3 4
Fig. 12-3
(a) B (b) M
Solutions
(a) (2, 2) (b) ( 3, 3)
(c) O (d) L
(e) N (f) G
(g) P (h) Q
(i) R (j) S
(c) (0, 0) (d) ( 4, 3)
(e) ( 11, 11) 2 2 (f) (11, 2 21) 2
(g) (0, 3) (h) ( 2, 0)
(i) (0,
11) 2
(j) (21, 0) 2
12.2 Coordinates of points in the four quadrants What are the signs of the coordinates of (a) a point in quadrant I; (b) a point in quadrant II; (c) a point in quadrant III; (d) a point in quadrant IV Show which coordinate has a sign and which zero value for a point between quadrants (e) IV and I; (f) I and II; (g) II and III; (h) III and IV.
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