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Loci and Construction

Loci and Construction. Objectives. Students are able to identify the locus of a set of points that are: at a given distance d from a given point O at a given distance d from a given straight line equidistant from two given points equidistant from two given intersecting straight lines

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Loci and Construction

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  1. Loci and Construction

  2. Objectives • Students are able to identify the locus of a set of points that are: • at a given distance d from a given point O • at a given distance d from a given straight line • equidistant from two given points • equidistant from two given intersecting straight lines • locus of a set of points that satisfy the above conditions using a compass, ruler and protractor • a triangle given any three sides/angles using a ruler, compass and protractor

  3. Objectives • Students are able to identify the locus of a set of points that are • >,<, ≥, ≤ a given distance d from a given point O • >,<, ≥, ≤ a given distance d from a given straight line • nearer to point A than point B • nearer to line A than line B

  4. Equidistant from two given points ( ( The locus is a perpendicular bisector of the line AB A B ( (

  5. At a given distance, d, from a given straight line d The locus is a pair of lines parallel to the given line, AB at a distance d cm from AB A B d

  6. Equidistant from two given intersecting lines ( ( ( The locus is the angle bisector of the angle between the two intersecting lines

  7. At a given distance, d, from a given point The locus is a circle with center A, and radius d cm. d X A

  8. To be at right angle to a given line, AB The locus is a circle with center AB as the diameter of the circle A B

  9. Example 1 • Describe the locus of a point P, which moves in a plane so that it is always 4cm from a fixed point O in the plane. The locus is a circle with center O, and radius 4cm. 4 cm X O

  10. Example 2 • Describe the locus of a point Q, which moves in a plane, so that it is always 5 cm from a given straight line, l. 5 cm The locus is a pair of lines parallel to the given line, l, at a distance 5 cm from it. l 5 cm

  11. Example 3 • Two points A and B are 7.5cm apart. Draw the locus of a point P, equidistant from A and B. ( ( The locus is a perpendicular bisector of the line AB A 7.5cm B ( (

  12. Example 4 • Draw two intersecting lines l and m. Draw the locus of a point P which moves such that it is equidistance from l and m. l ( ( The locus is the angle bisector of the angle between the two intersecting lines ( m

  13. Example 5 • Construct an angle XYZ equal to 60. Draw the locus of a point P, which moves such that it is equidistant from XY and YZ. X ( ( ( The locus is the angle bisector of the angle between the two intersecting lines 60 ( Z Y

  14. Example 6 • Construct the triangle ABC such that AB = 6cm,BC = 7cm and CA = 8cm. Draw the locus of P such that P is equidistant from A and C. ( ( C 8cm 7cm A 6cm B ( ( Locus of P

  15. Example 7 • Construct a triangle PQR in which QR = 8cm, angle RQP = 70 and segment RP = 9cm. Construct the locus which represents the points equidistant from PQ and QR. ( ( P 9cm ( 70 ( R 8cm Q Locus

  16. Example 8 • Constructing 60 angle Step 1: Construct Arc 1 Step 2: Construct Arc 2 Step 3: Draw line from intersection of two arc

  17. ( ( ( ( ( ( ( ( Example 9 • Construction of circumcircle Step 1: Draw perpendicular bisector of 1 side of triangle Step 2: Draw perpendicular bisector of 2nd side of triangle Step 3: Intersection of bisector will be the center of circle

  18. ( ( ( ( Example 10: • Construction of Inscribed Circle Step 1: Draw angle bisector on 1st angle of triangle Step 2: Draw angle bisector of 2nd angle of triangle Step 3: Intersection of angle bisector will be the center of circle ( (

  19. Independent Practice-1 • A long stick leans vertically against a wall. The stick then slides in such a way that its upper end describes a vertical straight line down the wall, while the lower end crosses the floor in a straight line at right angles to the wall. Construct a number of positions of the mid point of the stick and draw the locus.

  20. X Y 6cm Intersection of Loci • If two or more loci intersect at a point P, then P satisfies the conditions of the both loci simultaneously. • Example: ( ( • The circle is 6cm from point A. • The perpendicular bisector is at equidistant from point A and B. A B The point X and Y are both at : i) 6 cm from A ii) Equidistant from point A and B ( (

  21. Do it Yourself! Question 1 a) Construct and label triangle XYZ in which XY=10cm, YZ=7.5cm and angle XYZ = 60. Measure and write down the length of XZ. b) On your diagram, construct the locus of a point (i) 6cm from point Y (ii) equidistant from X and Z. c) The point P, inside the triangle XYZ is 6cm from Y and equidistant from point X and Y. (i) Label clearly, on your diagram, point P. (ii) Measure and write down the length of PX.

  22. Z X Y

  23. Do it Yourself! Question 5 A factory occupies a quadrilateral site ABCD in which AB=110m, BAD=65, AD=90m, ADC=110 and DC=60m. (a) Using a scale of 1cm to represent 10m, construct a plan of the quadrilateral ABCD. Measure ABC.

  24. Do it Yourself! (Continue) Two fuel storage tanks, T1 and T2 are located 30m from C and 15m from BD respectively. (b) On the same diagram, draw the locus which represents all the points inside the quadrilateral which are i) 30m from C ii) 15m from D (c) Mark clearly on your diagram, the positions of the tanks T1 and T2. (d) By measurement, find the distance between T1 and T2.

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