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CHAPTER 10 Regular Polygons and the Circle
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10.41. If the radius of a circle is 4, find the area of each segment formed by an inscribed (a) equilateral triangle; (b) regular hexagon; (c) square. (10.13 and 10.14) 10.42. Find the area of each segment of a circle if the segments are formed by an inscribed (a) Equilateral triangle and the radius of the circle is 6 (b) Regular hexagon and the radius of the circle is 3 (c) Square and the radius of the circle is 6 (10.13)
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Fig. 10-15
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10.43. Find the shaded area in each part of Fig. 10-15. Each heavy dot represents the center of an arc or a circle. (10.15) 10.44. Find the shaded area in each part of Fig. 10-16. Each dot represents the center of an arc or circle. (10.15)
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Fig. 10-16
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Locus
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11.1 Determining a Locus
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Locus, in Latin, means location. The plural is loci. A locus of points is the set of points, and only those points, that satisfy given conditions. Thus, the locus of points that are 1 in from a given point P is the set of points 1 in from P. These points lie on a circle with its center at P and a radius of 1 in, and hence this circle is the required locus (Fig. 11-1). Note that we show loci as long-short dashed figures.
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Fig. 11-1
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To determine a locus, (1) state what is given and the condition to be satisfied; (2) find several points satisfying the condition which indicate the shape of the locus; then (3) connect the points and describe the locus fully. All geometric constructions require the use of straightedges and compasses. Hence if a locus is to be constructed, such drawing instruments can be used.
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11.1A Fundamental Locus Theorems
PRINCIPLE 1:
The locus of points equidistant from two given points is the perpendicular bisector of the line segment joining the two points (Fig. 11-2).
Fig. 11-2
PRINCIPLE 2:
The locus of points equidistant from two given parallel lines is a line parallel to the two lines and midway between them (Fig. 11-3). The locus of points equidistant from the sides of a given angle is the bisector of the angle (Fig. 11-4).
PRINCIPLE 3:
CHAPTER 11 Locus
Fig. 11-3
PRINCIPLE 4:
Fig. 11-4
The locus of points equidistant from two given intersecting lines is the bisectors of the angles formed by the lines (Fig. 11-5).
Fig. 11-5
PRINCIPLE 5:
Fig. 11-6
The locus of points equidistant from two concentric circles is the circle concentric with the given circles and midway between them (Fig. 11-6). The locus of points at a given distance from a given point is a circle whose center is the given point and whose radius is the given distance (Fig. 11-7).
PRINCIPLE 6:
Fig. 11-7
PRINCIPLE 7:
Fig. 11-8
The locus of points at a given distance from a given line is a pair of lines, parallel to the given line and at the given distance from the given line (Fig. 11-8). The locus of points at a given distance from a given circle whose radius is greater than that distance is a pair of concentric circles, one on either side of the given circle and at the given distance from it (Fig. 11-9). The locus of points at a given distance from a given circle whose radius is less than the distance is a circle, outside the given circle and concentric with it (Fig. 11-10). (If r d, the locus also includes the center of the given circle.)
PRINCIPLE 8:
PRINCIPLE 9:
Fig. 11-9
Fig. 11-10
CHAPTER 11 Locus
SOLVED PROBLEMS
11.1 Determining loci Determine the locus of (a) a runner moving equidistant from the sides of a straight track; (b) a plane flying equidistant from two separated aircraft batteries; (c) a satellite 100 mi above the earth; (d) the furthermost point reached by a gun with a range of 10 mi.
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