Practice Makes Perfect 3 (Loci in Two Dimensions)

PRACTICE MAKES PERFECT: LOCI IN TWO DIMENSIONS PMR 2004 1. Diagram 1 in the answer space shows four squares, PKJN, KQLJ, HJMS and JLRM. W, X and Y are three moving points in the diagram. a) W moves such that it is equidistant from the straight lines PS and QR. By using the letters in the diagram, state the locus of W. b) On the diagram, draw i) the locus of X such that XJ = JN ii) the locus of Y such that its distance from point Q and point S are the same iii) Hence, mark with the symbol x all the intersections of the locus of X and the locus of Y. [5 marks] Answer : P K Q

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PMR 2005 2. Diagram 2 in the answer space shows four squares PQRS drawn on a grid of equal squares with sides of 1 unit. M, X and Y are three moving points in the diagram. a) M is the point which moves such that its distance from point Q and point S are the same. By using the letters in the diagram, state the locus of M.

ii)

b) On the diagram, draw i) the locus for the point X that is constantly 5 units from the line QR, the locus for the point Y that is constantly 7 units from the line R. c) Hence, mark with the symbol x all the intersections of the locus of X and the locus of Y. [5 marks] Answers. a) b) i), ii) c)

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PMR 2006 3. Diagram 3 in the answer space shows four squares PQRS drawn on a grid of equal squares with sides of 1 unit. W, X and Y are three moving points in the diagram a) W is the point which moves such that it is equidistant from the straight line PQ and PS. By using the letters in the diagram, state the locus of W. b) On the diagram, draw i) the locus for the point X that is constantly 5 units from the point P ii) the locus for the point Y that is constantly 4 units from the straight line QR. c) Hence, mark with the symbol x the intersections of the locus of X and the locus of Y. [5 marks] Answer : a) b) i), ii) c) P

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PMR 2007 (a) Diagram 5 shows a rhombus MNOP.

X is a moving point in the rhombus such that it is always equidistant from the straight lines PM and PO. By using the letters in the diagram, state the locus of X. (b) Diagram 6 in the answer space shows a regular hexagon ABCDEF. Y and Z are two moving points in the hexagon. On Diagram 6, draw

the locus of Y such that YD = DE, (ii) the locus of Z such that it is equidistant from point B and point F. (c) Hence, mark with the symbol x the intersection of the locus of Y and the locus of Z. [5 marks] Answer (a) (b) (i), (ii) (c)

5. Diagram 5 in the answer space shows two squares, JKNP and KLMN, each with sides 4 cm. X, V and Z are three moving points in the diagram. (a) X moves such that it is equidistant from the straight lines JP and LM. By using the letters in the diagram, state the locus of X. (b) On the diagram, draw (i) the locus of Y such that YN = 3 cm, (ii) the locus of Z such that ZP = ZJ. (c) Hence, mark with the symbol x for all the intersections of the locus of V and the locus of Z. [5 marks] Answer: (a) (b) (i), (ii) (c)

6. Diagram 6 in the answer space shows a right-angled triangle PQR. X, Y and Z are three moving points in the diagram (a) X moves such that it is equidistant from point P and point R. By using letters in the diagram, state the locus of X. (b) On the diagram, draw (i) the locus of Y such that PR = QR, (ii) the locus of Z such that it is always 4 cm from Q. (c) Hence, mark with the symbol x for all the intersections of the locus of V and the locus of Z. [5 marks] Answer: (a) (b) (i), (ii) (c)

7. Diagram 5 in the answer space shows four squares, AQPT QBRP TPSD and PRCS drawn on a grid of equal squares with sides 1 unit. V, W and X are three moving points in the diagram. (a) V moves such that it is equidistant from the straight lines AB and DC. By using the letters in the diagram, state the locus of V. (b) On the diagram, draw (i) the locus of point W that is constantly 4 units form the line AD, (ii) the locus of point X that is constantly 6 units from point D. (c) Hence, mark with the symbol x for all the intersections of the locus of W and the locus of X. [5 marks] Answer: (a) (b) (i), (ii) (c)

8. Diagram 7 in the answer space shows a rectangle JKLM drawn on a grid of equal squares with sides 1 unit. T, X and Y are three moving points in the diagram. (a) T is the point which moves such that its distance from point M and L are the same. By using the letters in the diagram, state the locus of T. (b) On the diagram, draw (i) the locus of X such that XL =7 units, (ii) the locus of Y such that the perpendicular distance of V from the line KL is always 4 units. (c) Hence, mark with the symbol x for all the intersections of the locus of X and the locus of V. [5 marks] Answer: (a) (b)

9. Diagram 5 in the answer space shows a rhombus PQRS with sides 4 cm. W, X and Y are three moving points in the diagram. (a) W is the point which moves such that its distance from point P and point R are the same. By using the letters in the diagram, state the locus of W. (b) On the diagram, draw (i) the locus of X such that XP = 3 cm, (ii) the locus of Y such that QY = RY. (c) Hence, mark with the symbol ® for all the intersections of the locus of X and the locus of V. [5 marks] Answer: (a) (b) (i), (ii), (c)

10. Diagram 6 in the answer space shows the rhombus PQRS drawn on a grid of equal squares with sides 1 unit. X, Y and Z are three moving points in the diagram. (a) X is the point which moves such that its distance from point Q and point S are the same. By using the letters in the diagram, state the locus of X. (b) On the diagram, draw (i) the locus of point V such that it is constantly 8 units from point 5, (ii) the locus of point Z such that it is constantly 5 units from the line PQ. (c) Hence, mark with the symbol ® for all the intersections of the locus of V and the locus of Z. [5 marks] Answer: (a) (b) (i), (ii), (c)

11. Diagram 6 in the answer space shows a regular hexagon PQRSTU of sides 3 cm. Y and Z are two moving points in the diagram. On the diagram, (a) construct the locus of Y such that QY = UY, (b) construct the locus of Z such that the distance of Z from RT is 3 cm, (c) mark with the symbol x for all the intersections of the locus of Y and the locus of Z. [5 marks] Answer: