Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK Preparing for the Regents Examination Geometry, AK
18 Two lines, one on each side, parallel to the given line and at a distance r from the given line 19 One line parallel to the two parallel lines and midway between them 20 The diameter of the circle that is the perpendicular bisector of the given chord 21 The diameter of the circle that is the perpendicular bisector of the given chord 22 A ray that is the angle bisector 23 A line perpendicular to the point on the given line 24 The perpendicular bisector of the chord that joins the two given points Locus in the Coordinate Plane (Pages 418–419) 1 (4) y 2 or y 2 2 (1) x 2 3 (2) x 5 4 (2) y 1 5 (4) (x 1) 2 (y 3) 2 25 6 (3) y 3 and x 0 7 (4) y x 1 and y x 5 8 y 1 9 y 2, y 10 10 y 4 11 y x 3 and y x 3 12 y x 3 13 x 1 and y 0 14 a (x 1) 2 (y 4) 2 16 b x 2 y 2 36 c x 2 (y 1) 2 9 d (x 1) 2 y 2 1 e x 2 (y 2) 2 6.25 f (x 1) 2 (y 3) 2 30.25 15 The showers could be placed anywhere on the perpendicular bisector of the line segment joining the diving boards, which is 35 ft from each of the diving boards. 14-5 Compound Locus and the Coordinate Plane (pages 421–422) 1 (3) a pair of points 2 (4) the empty set 3 (1) 0 4 (3) a pair of points 5 (3) 2 6 (2) 1 7 a 4 b 2 c 0 8 Find the point of concurrency (circumcenter) of the perpendicular bisectors of the sides of the triangle formed. 9 3 10 2 11 Parallel lines 2 inches from the given line, one on each side a Circle with radius 3, center R. 1 point b Circle with radius 6, center R. 2 points c Circle with radius 7, center R. 3 points d Circle with radius 9, center R. 3 points 12 For all parts, check students’ sketches. a Two intersecting circles, but not tangent. 2 points b Two tangent circles. 1 point c Two disjoint circles. 0 points 13 Sketch circle and two lines; 4 points. 14 Sketch a line and a circle; 2 points. 15 Sketch two lines bisecting vertical angles, and two parallel lines; 4 points. 16 Sketch two concentric circles and two lines bisecting vertical angles; 8 points. 17 The intersection of the line parallel to m and k and midway between them, and a circle with A as the center and d as the radius. Case I d 1 _ f (0 points) 2 Case II d 1 _ f (1 point) 2 Case III d 1 _ f (2 points) 2 18 The intersection of the perpendicular bisector of segment AB and a circle about center A with radius x. Case I Radius x 1 _ d (0 points) 2 Case II Radius x 1 _ d (1 point) 2 Case III Radius x 1 _ d (2 points) 2 Compound Loci and the Coordinate Plane (pages 423–424) 1 (4) 4 2 (3) 2 3 (4) 4 4 a 2 b 1 c 0 14-5 Compound Locus and the Coorinate Plane 89
5 Two horizontal lines y 11 and y 1, and vertical line x 4. Locus: 2 points. 6 Locus 2 points: (0, 8) and (8, 0) 7 Locus 2 points: (2, 7) and (2, 1) 8 Locus 2 points: (3, 1) and (2, 2) 9 a y 2 b x 4 c (4, 2) d (x 2) 2 (y 2) 2 16 e one 10 a Circle with radius, d b Two lines: x 1 and x 1 c (i) 1 point (ii) 3 points (iii) 4 points 14-6 Locus of Points Equidistant From a Point and a Line (page 427) Note: For exercises 1–10, check students’ sketches. 1 b (0, 2), (2, 3), (2, 3) x 2 c y _ 2 4 2 b (2, 1), (2, 1), (6, 1) 2 c y _ x x 8 _ 2 1 _ 2 3 b (2, 0), (2, 2), (6, 2) 2 c y x _ 1 8 _ x 2 1 _ 2 4 b (3, 0.5), (0, 1), (6, 1) x 2 c y _ x 1 6 5 b (1, 0), (3, 4), (3, 4) y 2 c x _ 1 8 6 b (3, 1.5), (6, 0), (0, 0) x 2 c y _ x 6 7 b (2, 2), (6, 4), (2, 4) x 2 c y _ 1 8 _ x 2 5 _ 2 8 b (1, 4), (4, 10), (4, 2) 2 y c x _ 2y 12 _ 7 3 _ 3 9 b (0, 0), (2, 4), (2, 4) y 2 c x _ 8 10 b (3, 1.5), (1, 2), (6, 2) x 2 c y _ x 2 6 90 Chapter 14: Locus and Constructions 14-7 Solving Other Linear- Quadratic and Quadratic- Quadratic Systems (page 430) Note: For exercises 1–20, check students’ sketches. 1 (2, 0), (2, 0) 2 (2 √ 2 , 0), (2 √ 2 , 0) 3 (0, 5), (4, 3) 4 (3, 2) 5 (2, 3), (2, 5) 6 (2, 1), (1, 2) 7 (1, 1), (1, 1) 8 (5, 27), (1, 5) 9 ( 5 _ , 6) , (2, 5) 3 10 (4, 0), (4, 0) 11 ( √ 2 , 0), ( √ 2 , 0) 12 (3, 2), (6, 1) 13 (2 √ 2 , √ 2 ), (2 √ 2 , √ 2 ), ( √ 2 , 2 √ 2 ), ( √ 2 , 2 √ 2 ) 14 (0.5, 1.25), (4, 3) 15 (1, 0) 16 (1, 3), (1, 3) 17 (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ) 18 (2, 1), (2, 1) 19 (3, 0) 20 (2, 4), (2, 0) Chapter Review (pages 430–433) 1 (2) acute triangles 2 (4) median to side −− AC 3 (2) AAS 4 (2) two circles 5 (2) X lies on the locus of points equidistant from R and S. 6 (2) y 2x 3 7 (3) (x 3) 2 (y 4) 2 36 8 (1) x 3 9 (3) x 5 10 (1) (0, 0) and (0, 8) 11 (4) (0, 2) and (4, 2) 12 (3) perpendicular bisectors of the sides of the triangle 13 (3) 4 14 (3) 3 15 (3) 2
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- Page 141 and 142: Practice Regents Examination One (p
5 Two horizontal lines y 11 and y 1, and<br />
vertical line x 4. Locus: 2 points.<br />
6 Locus 2 points: (0, 8) and (8, 0)<br />
7 Locus 2 points: (2, 7) and (2, 1)<br />
8 Locus 2 points: (3, 1) and (2, 2)<br />
9 a y 2 b x 4 c (4, 2)<br />
d (x 2) 2 (y 2) 2 16 e one<br />
10 a Circle with radius, d<br />
b Two lines: x 1 and x 1<br />
c (i) 1 point (ii) 3 points (iii) 4 points<br />
14-6 Locus of Points<br />
Equidistant From a Point<br />
and a Line<br />
(page 427)<br />
Note: For exercises 1–10, check students’<br />
sketches.<br />
1 b (0, 2), (2, 3), (2, 3)<br />
x 2<br />
c y _ 2<br />
4<br />
2 b (2, 1), (2, 1), (6, 1)<br />
2<br />
c y _ x<br />
<br />
x<br />
8 _ <br />
2 1 _<br />
2<br />
3 b (2, 0), (2, 2), (6, 2)<br />
2<br />
c y <br />
x<br />
<br />
_ <br />
1<br />
8 _ x <br />
2 1 _<br />
2<br />
4 b (3, 0.5), (0, 1), (6, 1)<br />
x 2<br />
c y _ x 1<br />
6<br />
5 b (1, 0), (3, 4), (3, 4)<br />
y 2<br />
c x _ 1<br />
8<br />
6 b (3, 1.5), (6, 0), (0, 0)<br />
x 2<br />
c y <br />
_ x<br />
6<br />
7 b (2, 2), (6, 4), (2, 4)<br />
x 2<br />
c y <br />
_ <br />
1<br />
8 _ x <br />
2 5 _<br />
2<br />
8 b (1, 4), (4, 10), (4, 2)<br />
2<br />
y<br />
c x _<br />
2y<br />
<br />
12 _ <br />
7<br />
3 _<br />
3<br />
9 b (0, 0), (2, 4), (2, 4)<br />
y 2<br />
c x _<br />
8<br />
10 b (3, 1.5), (1, 2), (6, 2)<br />
x 2<br />
c y _ x 2<br />
6<br />
90 Chapter 14: Locus and Constructions<br />
14-7 Solving O<strong>the</strong>r Linear-<br />
Quadratic and Quadratic-<br />
Quadratic Systems<br />
(page 430)<br />
Note: For exercises 1–20, check students’<br />
sketches.<br />
1 (2, 0), (2, 0)<br />
2 (2 √ 2 , 0), (2 √ 2 , 0)<br />
3 (0, 5), (4, 3)<br />
4 (3, 2)<br />
5 (2, 3), (2, 5)<br />
6 (2, 1), (1, 2)<br />
7 (1, 1), (1, 1)<br />
8 (5, 27), (1, 5)<br />
9 ( 5 _ , 6) , (2, 5)<br />
3<br />
10 (4, 0), (4, 0)<br />
11 ( √ 2 , 0), ( √ 2 , 0)<br />
12 (3, 2), (6, 1)<br />
13 (2 √ 2 , √ 2 ), (2 √ 2 , √ 2 ), ( √ 2 , 2 √ 2 ),<br />
( √ 2 , 2 √ 2 )<br />
14 (0.5, 1.25), (4, 3)<br />
15 (1, 0)<br />
16 (1, 3), (1, 3)<br />
17 (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ),<br />
(2 √ 2 , √ 5 )<br />
18 (2, 1), (2, 1)<br />
19 (3, 0)<br />
20 (2, 4), (2, 0)<br />
Chapter Review (pages 430–433)<br />
1 (2) acute triangles<br />
2 (4) median to side −−<br />
AC<br />
3 (2) AAS<br />
4 (2) two circles<br />
5 (2) X lies on <strong>the</strong> locus of points equidistant<br />
from R and S.<br />
6 (2) y 2x 3<br />
7 (3) (x 3) 2 (y 4) 2 36<br />
8 (1) x 3<br />
9 (3) x 5<br />
10 (1) (0, 0) and (0, 8)<br />
11 (4) (0, 2) and (4, 2)<br />
12 (3) perpendicular bisectors of <strong>the</strong> sides of <strong>the</strong><br />
triangle<br />
13 (3) 4<br />
14 (3) 3<br />
15 (3) 2
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