AQA GCSE Mathematics Paper 1 (Foundation), June 2025: Question 17

1 mark · Easy difficulty · Short Answer

Identify the correct statement regarding the length of a chord compared to the radius of a circle.

Practise this question

Question

Question 17 asks: 'A chord is drawn on a circle. Which statement is correct? Tick one box.' It provides four multiple-choice options: 'The chord must be longer than the radius.', 'The chord must be equal in length to the radius.', 'The chord must be shorter than the radius.', and 'The chord could be longer than, equal in length to or shorter than the radius.' Total 1 mark.

Mark scheme

Show the mark scheme Mark scheme table for question 17 showing the correct answer: 'The chord could be longer than, equal in length to or shorter than the radius.' which awards 1 B mark (B1).

How to answer it

Circle Definitions: Chords vs Radii

📋 What This Question Tests

This question assesses your understanding of fundamental circle definitions and geometry vocabulary:

  • Definition and properties of a chord (a straight line connecting any two points on the circumference).
  • Definition of a radius (distance from the centre to any point on the circumference).
  • How the length of a chord changes depending on its position relative to the centre of the circle.

Question 17

Multiple Choice (1 Mark)

✅ Correct Answer

Tick the 4th box:

The chord could be longer than, equal in length to or shorter than the radius.

Mark Scheme: 1 mark (B1) for identifying the correct statement.

💡 Key Knowledge

  • Chord: Any straight line segment joining two points on the circumference of a circle.
  • Diameter: The longest possible chord in a circle. It passes directly through the centre, and its length is 2 × radius .
  • Range of chord lengths: A chord's length can range anywhere strictly greater than 0 up to a maximum of the diameter ( 0 < length ≤ 2r ).

📐 Step-by-Step Geometric Proof

Let the radius of the circle be r. Consider where the chord is drawn:

  1. Shorter than the radius ( length < r ):
    Pick two points very close together on the circumference. The line joining them can be made arbitrarily small (e.g. 0.1r), which is much shorter than the radius.
  2. Equal to the radius ( length = r ):
    If you connect the centre of the circle to both ends of the chord, you form a triangle with two sides equal to r. If the angle at the centre is exactly 60°, an equilateral triangle is formed, meaning the chord length is exactly equal to r.
  3. Longer than the radius ( length > r ):
    If the chord passes through the centre, it is the diameter. Its length is 2r , which is twice the radius and therefore longer than the radius.
Visualizing this: Imagine a circle of radius 5 cm. A tiny chord sliced off the edge could be 1 cm long (shorter). A chord spanning a 60° arc is 5 cm long (equal). A chord passing through the centre is 10 cm long (longer). Hence, all three possibilities can happen!

🧠 Exam Technique

  • Watch out for absolute words: Options containing the word "must" imply that it is always true without exception.
  • Use counter-examples: If you can sketch a single case where a chord is shorter than the radius, any statement saying it "must be longer" or "must be equal" is immediately ruled out.
  • Sketching a quick rough circle with different lines drawn across it takes 5 seconds and guarantees the mark.

❌ Common Errors

  • Confusing chord with diameter: Thinking a chord always has to pass through the centre or must be longer than the radius.
  • Assuming a chord is fixed in size: Believing a chord has a set standard length.
  • Misreading the options: Overlooking the word "could" versus "must". The first three options are absolute (must), whereas the true property is variable (could).

Topics

Geometry and measures · 3.4.1 Properties and constructions

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.