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Loci and constructions practice questions with answers

Foundation and Higher · Grade 4 · Non-calculator

A second set of loci and constructions questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

More questions

Question 1 Easy

A point P is always 14 cm from a fixed point A. P moves to make a complete circle. What is the diameter of this circle in centimetres?

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Answer: 28

  1. P is always 14 cm from A, so the radius is 14 cm.
  2. Diameter = 2 × radius = 2 × 14.
  • Method (1): P is always 14 cm from A, so the radius is 14 cm.
  • Accuracy (1): Answer 28

Question 2 Easy

A and B are 14 cm apart. The perpendicular bisector of AB crosses AB at M. How far is M from A?

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Answer: 7

  1. M is the midpoint of AB, and AB is 14 cm long.
  2. 14 ÷ 2 = 7.
  • Method (1): M is the midpoint of AB, and AB is 14 cm long.
  • Accuracy (1): Answer 7

Question 3 Medium

Angle ABC is 100°. The angle bisector of ABC is drawn. What is the size of each of the two equal angles it makes?

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Answer: 50

  1. Angle ABC = 100° is split into two equal angles.
  2. 100° ÷ 2 = 50°.
  • Method (1): Angle ABC = 100° is split into two equal angles.
  • Accuracy (1): Answer 50

Question 4 Medium

A point P is always 13 cm from a fixed point A. P moves to make a complete circle. What is the diameter of this circle in centimetres?

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Answer: 26

  1. P is always 13 cm from A, so the radius is 13 cm.
  2. Diameter = 2 × radius = 2 × 13.
  • Method (1): P is always 13 cm from A, so the radius is 13 cm.
  • Accuracy (1): Answer 26

Question 5 Hard

A and B are 16 cm apart. P is on the perpendicular bisector of AB and is 6 cm from the midpoint M of AB. How far is P from A?

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Answer: 10

  1. M is the midpoint, so AM = 16 ÷ 2 = 8 cm.
  2. PA² = 8² + 6² = 64 + 36 = 100.
  • Method (1): M is the midpoint, so AM = 16 ÷ 2 = 8 cm.
  • Accuracy (1): Answer 10

Question 6 Hard

Angle ABC is 50°. The angle bisector of ABC is drawn. What is the size of each of the two equal angles it makes?

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Answer: 25

  1. Angle ABC = 50° is split into two equal angles.
  2. 50° ÷ 2 = 25°.
  • Method (1): Angle ABC = 50° is split into two equal angles.
  • Accuracy (1): Answer 25

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 13 marks

A and B are 24 cm apart. P is on the perpendicular bisector of AB and is 5 cm from the midpoint M of AB. How far is P from A?

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Answer: 13

  1. M is the midpoint, so AM = 24 ÷ 2 = 12 cm.
  2. PA² = 12² + 5² = 144 + 25 = 169.
  3. PA = √169 = 13 cm.
  • Method (1): M is the midpoint, so AM = 24 ÷ 2 = 12 cm.
  • Method (1): PA² = 12² + 5² = 144 + 25 = 169.
  • Accuracy (1): Answer 13

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

A point P is always 11 cm from a fixed point A. P moves to make a complete circle. What is the diameter of this circle in centimetres?

  1. 22
  2. 11
  3. 121
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22 The result is 22.

Check 2

A and B are 20 cm apart. The perpendicular bisector of AB crosses AB at M. How far is M from A?

  1. 10
  2. 20
  3. 5
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10 The result is 10.

Check 3

The point P is the same distance from A and from B. P is 13 cm from A. How far is P from B?

  1. 13
  2. 26
  3. 6.5
Show answer

13 The result is 13.

Check 4

A and B are 30 cm apart. P is on the perpendicular bisector of AB and is 8 cm from the midpoint M of AB. How far is P from A?

  1. 17
  2. 23
  3. 289
Show answer

17 The result is 17.

Check 5

Angle ABC is 30°. The angle bisector of ABC is drawn. What is the size of each of the two equal angles it makes?

  1. 15
  2. 30
  3. 165
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15 The result is 15.

Check 6

A point P is always 10 cm from a fixed point A. P moves to make a complete circle. What is the diameter of this circle in centimetres?

  1. 20
  2. 10
  3. 100
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20 The result is 20.

Check 7

A and B are 22 cm apart. The perpendicular bisector of AB crosses AB at M. How far is M from A?

  1. 11
  2. 22
  3. 5.5
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11 The result is 11.

Check 8

The point P is the same distance from A and from B. P is 7 cm from A. How far is P from B?

  1. 7
  2. 14
  3. 3.5
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7 The result is 7.

Check 9

A and B are 8 cm apart. P is on the perpendicular bisector of AB and is 3 cm from the midpoint M of AB. How far is P from A?

  1. 5
  2. 7
  3. 25
Show answer

5 The result is 5.

Check 10

Angle ABC is 60°. The angle bisector of ABC is drawn. What is the size of each of the two equal angles it makes?

  1. 30
  2. 60
  3. 150
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30 The result is 30.

Check 11

A point P is always 9 cm from a fixed point A. P moves to make a complete circle. What is the diameter of this circle in centimetres?

  1. 18
  2. 9
  3. 81
Show answer

18 The result is 18.

Check 12

A and B are 24 cm apart. The perpendicular bisector of AB crosses AB at M. How far is M from A?

  1. 12
  2. 24
  3. 6
Show answer

12 The result is 12.

Check 13

The point P is the same distance from A and from B. P is 14 cm from A. How far is P from B?

  1. 14
  2. 28
  3. 7
Show answer

14 The result is 14.

Check 14

A and B are 10 cm apart. P is on the perpendicular bisector of AB and is 12 cm from the midpoint M of AB. How far is P from A?

  1. 13
  2. 17
  3. 169
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13 The result is 13.

Check 15

Angle ABC is 90°. The angle bisector of ABC is drawn. What is the size of each of the two equal angles it makes?

  1. 45
  2. 90
  3. 135
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45 The result is 45.

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