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CHAPTER 11 Locus
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Then prove that any point satisfying the condition is on the locus: Given: Any point Q which is equidistant from points A and B (QA > QB). To Prove: Q is on the perpendicular bisector of AB. Plan: Draw QG perpendicular to AB and prove by congruent triangles that QG bisects AB. PROOF:
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Statements 1. Draw QG ' AB . 2. QA > QB 3. /QGA and /QGB are rt. /s; ^QGA and ^QGB are rt. ^s. 4. QG > QG 5. ^QGA > ^QGB 6. AG > GB 7. QG bisects AB . 8. QG is ' bisector of AB.
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Reasons 1. Through an external point, a line can be drawn perpendicular to a given line. 2. Given 3. Perpendiculars form right angles; ^s with a rt./ are rt. ^s. 4. Reflexive property 5. Hy-leg 6. Corresponding parts of > triangles are >. 7. To bisect is to divide into two congruent parts 8. A line perpendicular to a segment and bisecting it is its perpendicular bisector.
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11.1. Determine the locus of (a) The midpoints of the radii of a given circle (b) The midpoints of chords of a given circle parallel to a given line (c) The midpoints of chords of fixed length in a given circle (d) The vertex of the right angle of a triangle having a given hypotenuse (e) The vertex of an isosceles triangle having a given base (f) The center of a circle which passes through two given points (g) The center of a circle tangent to a given line at a given point on that line (h) The center of a circle tangent to the sides of a given angle 11.2. Determine the locus of (a) A boat moving so that it is equidistant from the parallel banks of a stream (b) A swimmer maintaining the same distance from two floats (c) A police helicopter in pursuit of a car which has just passed the junction of two straight roads and which may be on either one of them (d) A treasure buried at the same distance from two intersecting straight roads 11.3. Determine the locus of (a) a planet moving at a fixed distance from its sun; (b) a boat moving at a fixed distance from the coast of a circular island; (c) plants laid at a distance of 20 ft from a straight row of other plants; (d) the outer extremity of a clock hand. (11.1) (11.1) (11.1)
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CHAPTER 11 Locus
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11.4. Excluding points lying outside rectangle ABCD in Fig. 11-15, find the locus of points which are (a) Equidistant from AD and BC (b) Equidistant from AB and CD (c) Equidistant from A and B (d) Equidistant from B and C (e) 5 units from BC (f) 10 units from AB (g) 20 units from CD (h) 10 units from B
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(11.1)
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Fig. 11-15
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Fig. 11-16
11.5. Find the locus of points in rhombus ABCD in Fig. 11-16, which are equidistant from (a) AB and AD; (b) AB and (11.1) BC; (c) A and C; (d) B and D; (e) each of the four sides. 11.6. In Fig. 11-17, find the locus of points which are on or inside circle C and (a) 5 units from O (b) 15 units from O (c) Equidistant from circles A and C (d) 10 units from circle C (e) 10 units from circle A (f) 5 units from circle B (g) The center of a circle tangent to circles A and C (11.1 and 11.2)
Fig. 11-17
11.7. Determine the locus of the center of (a) a coin rolling around and touching a smaller coin; (b) a coin rolling around and touching a larger coin; (c) a wheel moving between two parallel bars and touching both of them; (d) a wheel moving along a straight metal bar and touching it. (11.2) 11.8. Find the locus of points that are in rectangle ABCD of Fig. 11-18 and the center of a circle (a) Tangent to AD and BC (b) Tangent to AB and CD (c) Tangent to AD and EF (d) Of radius 10, tangent to BC (e) Of radius 20, tangent to AD (f) Tangent to BC at G (11.2)