1. 1
   A landscape architect is designing a triangular flower bed and marks two points, P and Q, on the ground to represent two corners of the bed, 6 metres apart. She plans to use compass and straightedge constructions to mark out other key features of the design accurately, without using a measuring tape.

   (a)

   Which of the following correctly describes every point that lies on the perpendicular bisector of the line segment PQ?

   [1 mark]

   • AIt is equidistant from P and Q
   • BIt is exactly 3 m from P only
   • CIt lies exactly halfway along PQ only
   • DIt is equidistant from every point on the ground

   (b)

   The architect also needs to bisect an angle at one corner of the flower bed. Which statement correctly describes every point on the bisector of an angle?

   [1 mark]

   • AIt is the same distance from the vertex as the length of one arm
   • BIt is equidistant from the two lines (arms) forming the angle
   • CIt divides the angle into two unequal parts
   • DIt is always perpendicular to both arms of the angle

   (c)

   When constructing the perpendicular bisector of segment PQ using compasses, the architect draws an arc from P and an arc from Q on each side of the line. What is required for these arcs to intersect correctly and give a true perpendicular bisector?

   [1 mark]

   • AAny compass width may be used, as long as both arcs are drawn from the same point
   • BThe compass width for the arc from P must be exactly half of PQ, and for Q it must be the full length of PQ
   • CBoth arcs must be drawn using the same compass width, which must be more than half the length of PQ
   • DThe compass width must be less than a quarter of the length of PQ

   Total for question 1: 3 marks

2. 2
   Two mobile phone masts, mast A and mast B, stand 8 km apart in flat countryside. The network provider wants to build a new mast, mast C, that receives an equally strong signal path from both A and B, so it must be positioned at exactly the same distance from mast A as from mast B.

   (a)

   Describe fully the locus of all possible positions for mast C, and state where this locus is positioned relative to masts A and B.

   [2 marks]

   (b)

   A second mast, mast D, must also be equidistant from A and B, but it must additionally be more than 5 km from mast A. Explain how you would identify the set of all possible positions for mast D, referring to the loci involved.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A surveyor is asked to construct a triangular plot of land, triangle XYZ, on paper before pegging it out on site, where XY = 9 cm, XZ = 7 cm and the included angle YXZ = 50 degrees.

   (a)

   Explain, step by step, how the surveyor would use a ruler, compasses and a protractor to construct triangle XYZ accurately from these measurements, and explain why this method guarantees that only one triangle is possible.

   [3 marks]

   (b)

   A second surveyor claims that triangle XYZ could equally well be constructed if she were told only the three angles of the triangle, without any side lengths. Explain, with reference to the properties of triangle construction, why this claim is incorrect, and state what additional information would be needed to fix the triangle's size.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A farmer owns a rectangular field ABCD with AB = 60 m and BC = 40 m, where AB and CD are the longer, parallel sides. A single water sprinkler must be installed inside the field, positioned so that it is at least 15 m from vertex A, no more than 25 m from vertex C, and closer to side AB than to side CD.

   (a)

   Describe fully the locus of points that are exactly 15 m from vertex A, and explain how this locus separates the field into a region where the 'at least 15 m from A' condition is satisfied and a region where it is not.

   [4 marks]

   (b)

   Describe fully the locus of points that are equidistant from side AB and side CD, and use this locus to determine which points in the field are closer to AB than to CD.

   [4 marks]

   (c)

   Using your answers to parts (a) and (b), and given that the sprinkler must also be no more than 25 m from vertex C, describe fully the region within the field where the sprinkler can be installed so that all three conditions are satisfied simultaneously. State whether this region is non-empty, and justify your answer by referring to the boundaries you have identified.

   [5 marks]

   Total for question 4: 13 marks

End of questions