MCQs / Mathematics / Loci

Loci MCQs — Mathematics

23 multiple-choice questions on Loci from Mathematics, each with the correct answer and, where available, a teacher's explanation. Read through them to revise the chapter, then practise the same questions one by one to test yourself.

1. The locus of points at a fixed distance d from a straight line is:

  1. a pair of lines parallel to it, one on each side, at distance d
  2. one perpendicular line
  3. a circle
  4. a single parallel line

Answer: A) a pair of lines parallel to it, one on each side, at distance d

Explanation: A pair of lines parallel to it, one on each side, at distance d. SLO M-09-B-19 · Application

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2. A new water tank must be equally far from two villages A and B. It must be built on:

  1. the line through A and B
  2. the perpendicular bisector of AB
  3. a circle centred at A
  4. the angle bisector at A

Answer: B) the perpendicular bisector of AB

Explanation: Points equidistant from A and B lie on the perpendicular bisector of AB. SLO M-09-B-19 · Application

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3. A circular race track has inner radius 30 m and outer radius 35 m. The track is the locus of points that are:

  1. more than 35 m from the centre
  2. exactly 35 m from the centre
  3. exactly 30 m from the centre
  4. at least 30 m and at most 35 m from the centre

Answer: D) at least 30 m and at most 35 m from the centre

Explanation: It is the ring between the two circles. SLO M-09-B-19 · Application

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4. AB = 8 cm. How many points are exactly 3 cm from A and 3 cm from B?

  1. 1
  2. 0
  3. 2
  4. 4

Answer: B) 0

Explanation: Draw circles of radius 3 about A and B. Since AB = 8 is greater than 2 3 = 6, the circles meet in 0 points. SLO M-09-B-19 · Application

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5. A path must stay the same distance from two straight walls that meet at a corner. It follows:

  1. the bisector of the angle between the walls
  2. the perpendicular bisector of the walls
  3. a circle around the corner
  4. a line parallel to one wall

Answer: A) the bisector of the angle between the walls

Explanation: Points equidistant from two intersecting lines lie on the angle bisector. SLO M-09-B-19 · Application

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6. AB = 7 cm. How many points are exactly 3.5 cm from A and 3.5 cm from B?

  1. 0
  2. 4
  3. 2
  4. 1

Answer: D) 1

Explanation: Draw circles of radius 3.5 about A and B. Since AB = 7 is equal to 2 3.5 = 7, the circles meet in 1 point. SLO M-09-B-19 · Application

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7. AB = 10 cm. How many points are exactly 6 cm from A and 6 cm from B?

  1. 4
  2. 1
  3. 2
  4. 0

Answer: C) 2

Explanation: Draw circles of radius 6 about A and B. Since AB = 10 is less than 2 6 = 12, the circles meet in 2 points. SLO M-09-B-19 · Application

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8. A mobile tower must be within 10 km of town A and within 10 km of town B, and AB = 16 km. The possible positions form:

  1. a straight line
  2. a single point
  3. no region at all
  4. the overlap of two circular regions of radius 10 km

Answer: D) the overlap of two circular regions of radius 10 km

Explanation: 16 < 10 + 10, so the two discs overlap; any point in the overlap works. SLO M-09-B-19 · Application

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9. The locus of points equidistant from two parallel lines is:

  1. the perpendicular bisector
  2. two more parallel lines
  3. a circle
  4. a line parallel to both, midway between them

Answer: D) a line parallel to both, midway between them

Explanation: A line parallel to both, midway between them. SLO M-09-B-19 · Application

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10. Point P is 3 cm from a straight line ell. How many points on ell are 5 cm from P?

  1. 1
  2. 2
  3. 3
  4. 0

Answer: B) 2

Explanation: The circle of radius 5 about P cuts the line, because 3 < 5. SLO M-09-B-19 · Application

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11. AB = 5 cm. How many points are exactly 2 cm from A and 2 cm from B?

  1. 2
  2. 4
  3. 1
  4. 0

Answer: D) 0

Explanation: Draw circles of radius 2 about A and B. Since AB = 5 is greater than 2 2 = 4, the circles meet in 0 points. SLO M-09-B-19 · Application

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12. The locus of points at a fixed distance r from a fixed point O is:

  1. two parallel lines
  2. the perpendicular bisector
  3. a straight line through O
  4. a circle with centre O and radius r

Answer: D) a circle with centre O and radius r

Explanation: A circle with centre O and radius r. SLO M-09-B-19 · Application

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13. The point equidistant from three non-collinear points A, B, C is:

  1. the circumcentre — where the perpendicular bisectors of AB, BC, CA meet
  2. the centroid
  3. the midpoint of AB
  4. any point on AB

Answer: A) the circumcentre — where the perpendicular bisectors of AB, BC, CA meet

Explanation: The circumcentre — where the perpendicular bisectors of AB, BC, CA meet. SLO M-09-B-19 · Application

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14. AB = 6 cm. How many points are exactly 3 cm from A and 3 cm from B?

  1. 1
  2. 4
  3. 2
  4. 0

Answer: A) 1

Explanation: Draw circles of radius 3 about A and B. Since AB = 6 is equal to 2 3 = 6, the circles meet in 1 point. SLO M-09-B-19 · Application

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15. The locus of points equidistant from two fixed points A and B is:

  1. the perpendicular bisector of AB
  2. a circle with diameter AB
  3. the line AB
  4. the midpoint of AB only

Answer: A) the perpendicular bisector of AB

Explanation: The perpendicular bisector of AB. SLO M-09-B-19 · Application

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16. AB = 4 cm. How many points are exactly 3 cm from A and 3 cm from B?

  1. 0
  2. 4
  3. 2
  4. 1

Answer: C) 2

Explanation: Draw circles of radius 3 about A and B. Since AB = 4 is less than 2 3 = 6, the circles meet in 2 points. SLO M-09-B-19 · Application

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17. A treasure is 4 m from tree T and equally far from posts A and B. To find it you draw:

  1. a circle of radius 4 m about T and the perpendicular bisector of AB, and look where they cross
  2. the line AB only
  3. only the circle about T
  4. the angle bisector at T

Answer: A) a circle of radius 4 m about T and the perpendicular bisector of AB, and look where they cross

Explanation: The treasure lies on both loci, so it is at their intersection (0, 1 or 2 possible points). SLO M-09-B-19 · Application

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18. A sprinkler waters all points within 2.5 m of it. It is placed at the centre of a 4 m × 4 m square lawn. Which part of the lawn stays dry?

  1. the middle of each side
  2. none of it
  3. the four corners
  4. the centre

Answer: C) the four corners

Explanation: The middle of each side is 2 m from the centre (watered), but each corner is 2^2 + 2^2 = 2 2 approx 2.83 m away, which is more than 2.5 m. SLO M-09-B-19 · Application

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19. A goat tied by a 7 m rope can graze an area of about: (use pi approx 227)

  1. 154 m²
  2. 49 m²
  3. 44 m²
  4. 308 m²

Answer: A) 154 m²

Explanation: pi r^2 = 227 49 = 154 m². SLO M-09-B-19 · Application

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20. A goat is tied to a peg by a 5 m rope in a large field. The region it can graze is:

  1. a semicircle of radius 5 m
  2. a line 5 m long
  3. a circle (disc) of radius 5 m centred at the peg
  4. a square of side 5 m

Answer: C) a circle (disc) of radius 5 m centred at the peg

Explanation: Every point within 5 m of the peg can be reached. SLO M-09-B-19 · Application

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21. Point P is 5 cm from a straight line ell. How many points on ell are 5 cm from P?

  1. 2
  2. 0
  3. 3
  4. 1

Answer: D) 1

Explanation: The circle of radius 5 about P touches the line, because 5 = 5. SLO M-09-B-19 · Application

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22. The locus of points equidistant from two intersecting straight lines is:

  1. a line parallel to both
  2. the pair of lines bisecting the angles between them
  3. the point of intersection only
  4. a circle

Answer: B) the pair of lines bisecting the angles between them

Explanation: The pair of lines bisecting the angles between them. SLO M-09-B-19 · Application

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23. Point P is 7 cm from a straight line ell. How many points on ell are 5 cm from P?

  1. 0
  2. 3
  3. 2
  4. 1

Answer: A) 0

Explanation: The circle of radius 5 about P does not reach the line, because 7 5. SLO M-09-B-19 · Application

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