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GCSE · Geometry and Measures

4.15 Loci and Construction

EDEXCEL 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Loci vocabulary. Complete parts (a) to (c).

(a)State the name given to the locus of a point that stays a fixed distance from a single fixed point.(1)

(b)State the name given to the locus of a point that is always the same distance from two fixed points, A and B.(1)

(c)State the name given to the locus of a point that is always the same distance from two straight lines that cross each other.(1)

(Total for Question 1 is 3 marks)

Draw a straight line segment AB of length 7 cm. Using a ruler and compasses only, construct the perpendicular bisector of AB. You must show all your construction lines.

(Total for Question 2 is 2 marks)

A road planner needs to bisect an angle at a junction.

(a)Draw an angle ABC = 76 degrees, with BA and BC each about 6 cm long.(1)

(b)Using a ruler and compasses only, construct the bisector of angle ABC. Show all your construction lines.(2)

(Total for Question 3 is 3 marks)

Draw a straight horizontal line segment XY, 9 cm long. Mark a point P above the line, not on it. Using a ruler and compasses only, construct the perpendicular line from P to the line XY. Show all your construction lines.

(Total for Question 4 is 2 marks)

Draw a straight line 10 cm long and mark a point M on the line, roughly in the middle. Using a ruler and compasses only, construct the perpendicular to the line at the point M. Show all your construction lines.

(Total for Question 5 is 2 marks)

Mark a point P on a straight line. Using only a ruler and compasses (no protractor), construct an angle of 60 degrees at P.

(Total for Question 6 is 2 marks)

A garden sprinkler is placed at point S. It waters the ground up to a distance of 3.5 m from S. Using a scale of 1 cm to 1 m, draw the locus of the boundary of the area that gets watered.

(Total for Question 7 is 2 marks)

Two mobile phone masts, A and B, are 6 km apart. Using a scale of 1 cm to 1 km, draw the positions of A and B. Then, using a ruler and compasses only, construct the locus of points that are equidistant from both masts.

(Total for Question 8 is 3 marks)

Two straight roads meet at point O at an angle of 50 degrees. Draw the two roads (each about 7 cm long) meeting at O with this angle between them. A new footpath is to be built so that every point on it is the same distance from both roads. Using a ruler and compasses only, construct the locus of the footpath.

(Total for Question 9 is 3 marks)

A wall is represented by the straight line segment PQ, 6 cm long, using a scale of 1 cm to 2 m. Draw PQ. A safety zone extends exactly 3 m from every point on the wall, including its two ends. Using the same scale, draw the locus of the boundary of the safety zone.

(Total for Question 10 is 3 marks)

The incentre of a triangle is the point equidistant from all three sides, found by constructing the bisectors of all three interior angles. Explain why this point must always lie inside the triangle, for any triangle.

(Total for Question 11 is 2 marks)

Triangle construction using ruler and compasses only.

(a)Construct triangle ABC with AB = 6 cm, BC = 5 cm and CA = 4 cm. Show all construction arcs.(3)

(b)Measure the size of angle BAC.(1)

(Total for Question 12 is 4 marks)

A rectangular garden PQRS has PQ = 10 m and QR = 6 m, with P, Q, R, S labelled in order going around the rectangle. Using a scale of 1 cm to 1 m, draw the rectangle. A tree is to be planted so that it is (i) within 7 m of corner P, and (ii) nearer to side QR than to side PS. Construct the necessary loci and shade the region of the garden where the tree could be planted.

(Total for Question 13 is 4 marks)

A goat is tied by a rope of length 5 m to a metal stake in the middle of a large, flat field with no obstacles nearby.

(a)Using a scale of 1 cm to 1 m, draw the locus of the points the goat can reach.(2)

(b)Calculate the area of grass the goat can graze, giving your answer to 3 significant figures.(2)

(Total for Question 14 is 4 marks)

A triangular field ABC has AB = 90 m, BC = 70 m and angle ABC = 65 degrees. A fence post is to be placed so that it is (i) equidistant from sides AB and BC, and (ii) exactly 60 m from vertex A.

(a)Using a scale of 1 cm to 10 m, construct triangle ABC accurately.(1)

(b)Using a ruler and compasses only, construct the locus of points equidistant from AB and BC, and the locus of points 60 m from A. Hence find and label the position(s) of the fence post, X (and Y if there is more than one position).(4)

(Total for Question 15 is 5 marks)

A goat is tied by a rope of length 6 m to a corner of a rectangular shed that measures 5 m by 3 m, standing in an open field.

(a)Sketch and describe the locus of the points the goat can reach, taking into account the shed blocking part of the rope's swing.(2)

(b)Calculate the total area the goat can graze, giving your answer to 3 significant figures.(3)

(Total for Question 16 is 5 marks)

A radio transmitter is at point T, on a bearing of 060 degrees from a port P, and 20 km from P. The transmitter's signal can be picked up up to 16 km away. A ship sails from P on a bearing of 100 degrees. Use a scale of 1 cm to 4 km.

(a)Draw the positions of P and T, the ship's path from P, and the boundary of the transmitter's range.(3)

(b)Use your diagram to find the distance, to the nearest km, over which the ship is within range of the transmitter.(1)

(Total for Question 17 is 4 marks)

Mark scheme · 4.15 Loci and Construction

Question 1

(a) B1 a circle (centred on the fixed point), oe
(a) Answer: A circle.
(b) B1 the perpendicular bisector of the line segment AB, oe
(b) Answer: The perpendicular bisector of AB.
(c) B1 the angle bisector(s) of the angles between the lines, oe
(c) Answer: The bisector(s) of the angle(s) between the two lines.

Question 2

C1 two pairs of intersecting arcs of equal radius (radius greater than half of AB), one pair centred on A and one pair centred on B
A1 correct straight line drawn through both points of intersection, extending beyond AB, within 2 mm and 2 degrees of the true perpendicular bisector
Answer: A straight line perpendicular to AB, passing through its midpoint, 3.5 cm from both A and B.

Question 3

(a) B1 angle drawn 76 degrees, within 2 degrees, with both arms at least 5 cm
(a) Answer: Angle ABC = 76 degrees drawn with a protractor.
(b) C1 one arc, centred at B, of any radius, crossing both arms BA and BC
(b) A1 two further equal-radius arcs from those crossing points, intersecting each other, with the bisector ray drawn from B through this intersection, within 2 degrees of 38 degrees on each side
(b) Answer: The bisector splits angle ABC into two 38 degree angles.

Question 4

C1 one arc centred at P crossing the line XY at two points
A1 correct perpendicular line drawn through P, meeting XY at 90 degrees (within 2 degrees), using equal arcs from the two crossing points
Answer: A straight line through P that meets XY at a right angle (90 degrees).

Question 5

C1 two arcs of equal radius, centred at M, cutting the line on each side of M
A1 correct perpendicular line drawn through M using two further equal arcs from those points, within 2 degrees of 90 degrees to the original line
Answer: A straight line through M at 90 degrees to the original line.

Question 6

C1 arc centred at P crossing the line at a point Q, then a further arc of the same radius centred at Q, crossing the first arc at R
A1 line PR drawn correctly, with angle QPR measuring 60 degrees within 2 degrees, no protractor marks used
Answer: Angle QPR = 60 degrees, constructed using the equilateral-triangle compass method.

Question 7

B1 point S clearly marked
C1 circle of radius 3.5 cm (representing 3.5 m) drawn accurately centred on S
Answer: A circle of radius 3.5 cm (representing 3.5 m) centred on S.

Question 8

B1 A and B plotted correctly, 6 cm apart, to the given scale
C1 two pairs of intersecting arcs of equal radius (greater than 3 cm), centred on A and on B
A1 correct perpendicular bisector line drawn through the intersections, extending beyond the line AB
Answer: The perpendicular bisector of AB, crossing AB at its midpoint (3 km from each mast).

Question 9

B1 angle of 50 degrees drawn at O, within 2 degrees, with arms at least 6 cm long
C1 equal-radius arcs from O crossing both roads, followed by equal-radius arcs from those crossing points intersecting each other
A1 bisector ray drawn correctly from O through the intersection, splitting the 50 degree angle into two 25 degree angles (within 2 degrees)
Answer: The bisector of the 50 degree angle at O, giving a footpath at 25 degrees from each road.

Question 10

C1 two straight lines parallel to PQ, each 1.5 cm (3 m) from PQ, one on each side, matching the length of PQ
C1 two semicircular ends of radius 1.5 cm (3 m), centred on P and on Q, joining the parallel lines
A1 complete continuous locus with no gaps, forming a single closed stadium (racetrack) shape
Answer: A stadium (racetrack) shape: two straight lines 1.5 cm (3 m) either side of PQ, joined by semicircles of radius 1.5 cm (3 m) at each end.

Question 11

B1 each angle bisector lies between the two sides that form the interior angle it bisects, so it stays within the triangle's interior region near that vertex
B1 since all three bisectors lie in the region common to all three interior angles, their point of intersection must lie inside the triangle
Answer: Because each bisector is confined to the interior of its own angle, the point where all three meet must lie inside the triangle.

Question 12

(a) C1 line AB drawn, length 6 cm (within 2 mm)
(a) C1 arc of radius 5 cm centred on B and arc of radius 4 cm centred on A, intersecting at C
(a) A1 triangle ABC completed with all three sides correct within 2 mm and construction arcs left visible
(a) Answer: Triangle ABC with AB = 6 cm, BC = 5 cm, CA = 4 cm, constructed from two intersecting arcs.
(b) B1 angle BAC = 55.8 degrees, accept 54 to 58 degrees by measurement
(b) Answer: Angle BAC = 55.8 degrees (1 dp); accept 54 to 58 degrees from measurement.

Question 13

C1 rectangle PQRS drawn accurately to scale, 10 cm by 6 cm, correctly labelled
C1 circle (or arc) of radius 7 cm centred on P drawn accurately
C1 the line equidistant from QR and PS constructed correctly (a straight line parallel to QR and PS, 5 cm from each, using the perpendicular bisector method on a segment of PQ)
B1 correct region shaded, satisfying both conditions (inside the circle radius 7 cm centred on P, and on the QR side of the midline), cao
Answer: The shaded region is the part of the garden inside the circle of radius 7 m centred on P that also lies on the QR side of the line 5 m from P (parallel to QR and PS).

Question 14

(a) B1 stake position clearly marked
(a) C1 circle of radius 5 cm (5 m) drawn accurately, centred on the stake
(a) Answer: A circle of radius 5 cm (5 m) centred on the stake.
(b) M1 area = π x 5² oe
(b) A1 78.5 m², awrt
(b) Answer: 78.5 m² (3 s.f.)

Question 15

(a) C1 triangle ABC constructed with AB = 9 cm, angle ABC = 65 degrees (within 2 degrees), BC = 7 cm, all within 2 mm
(a) Answer: Triangle ABC drawn with AB = 9 cm, angle B = 65 degrees, BC = 7 cm.
(b) C1 angle bisector of angle ABC constructed correctly using compass arcs, drawn as a ray extended well beyond the triangle
(b) C1 arc of radius 6 cm (60 m) centred on A drawn, crossing the bisector ray
(b) B1 both intersection points located correctly (approximately 4.0 cm and 11.1 cm from B along the bisector, within construction tolerance) and labelled X and Y
(b) B1 correct conclusion that two positions exist, because the arc crosses the bisector ray at two points
(b) Answer: Two positions satisfy both conditions: approximately 40 m and 111 m from B along the bisector of angle B (accept 38 to 43 m and 108 to 115 m by construction tolerance); both lie exactly 60 m from A.

Question 16

(a) C1 a three-quarter circle of radius 6 m about the tie point, blocked by the shed over the remaining quarter turn
(a) C1 two additional quarter circles where the rope wraps around the adjacent corners of the shed, of radius 1 m (6 - 5) and radius 3 m (6 - 3)
(a) Answer: A three-quarter circle of radius 6 m, plus a quarter circle of radius 1 m wrapping round one adjacent corner and a quarter circle of radius 3 m wrapping round the other.
(b) M1 (3/4) x π x 6² oe, = 27pi
(b) M1 (1/4) x π x 1² + (1/4) x π x 3² oe, = 2.5pi
(b) A1 92.7 m², awrt
(b) Answer: 92.7 m² (3 s.f.)

Question 17

(a) C1 P and T plotted correctly, 5 cm apart, with T on a bearing of 060 degrees from P (within 2 degrees)
(a) C1 ship's path drawn as a ray from P on a bearing of 100 degrees (within 2 degrees), at least 10 cm long
(a) C1 circle of radius 4 cm (16 km) drawn accurately, centred on T
(a) Answer: P and T plotted 5 cm apart on a bearing of 060 degrees; the ship's path drawn on a bearing of 100 degrees from P; a circle of radius 4 cm centred on T.
(b) B1 answer awrt 19 km (accept 17 to 21 km, allowing for scale-drawing tolerance)
(b) Answer: Approximately 19 km (accept 17 to 21 km from an accurate scale drawing).

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