Circles

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AS Maths – Pure – Midpoints

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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6A Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

No Live Lesson by Bicen Maths for this Section

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(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 1-6✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 265b: Find the midpoint of two points given as coordinates.
• 265c: Find a coordinate given another and their midpoint.
• 265d: Solve problems involving the midpoint of two points where some values are unknown.
• 266a: Find the coordinates of the middle of three points in a line given the ratio of the distances between them.
• 266b: Find the coordinates of the last of three points in a line given the ratio of the distances between them.
• 266c: Find the coordinates of one of three points on a line given a multiplicative relationship for the distances between them.
• 266d: Find the coordinates of one of three points in a line given a ratio of the distances between two points and the whole line segment.
• 266e: Solve problems involving two points divided into a ratio where some values are unknown.
• 269c: Determine the midpoint of the line segment between the x and y intercepts of a straight line.
• 269e: Find the coordinate of a point between the x and y intercepts in a particular ratio.
• 265: Exam Practice: Midpoint of two points
• 266: Exam Practice: Splitting a line segment in a ratio
• 269: Exam Practice: x and y intercepts of a line

Other Videos

Finding the midpoint of a line segment

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• JellyMaths Coordinate Geometry Group Cards✦ (TES Account Required)
• The Chalk Face Midpoints, Line Segments

Worksheets

• The Chalk Face Midpoints Questions
• kutasoftware Worksheets
  • The Midpoint Formula
  • Using the Midpoint Formula

AS Maths – Pure – Perpendiculars

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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6B Slides (Courtesy of Owen134866)

Bicen Maths Live Lesson

Circles 1 • Midpoints and Perpendicular Bisectors • P1 Ex6A/B • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 1-6✦

Extra Questions

• Dr Frost Questions (Short)

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 445l: Determine the equation of the perpendicular bisector of two points, or the diagonal of a kite given the other.
• 495e: Determine the perpendicular bisector of a line using y-y₁=m(x-x₁)
• 445: Exam Practice: Equations of perpendicular lines
• 495: Exam Practice: Determining the equation of a straight line in point-slope form (y-y₁ = m(x-x₁))

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Standards Unit Connecting Perpendicular Lines✦

Worksheets

• Solomon C1 Co-Ordinate Geometry D Questions & Solutions
• Mathshelper (Jon Stone)
  • Perpendicular Bisector
• MadAsMaths (T. Madas)
  • Line Coordinate Geometry Exam Questions Standard Q28; Hard Q4 (difficulty 3–4 of 5)

AS Maths – Pure – Equation of a Circle

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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6C Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (Up to 9:51)
• 351 Find an equation of a circle centred at the origin which is drawn. Key Skill Video (GCSE)
• 353 Find an equation of a circle centred at the origin given its radius. Key Skill Video (GCSE)

Bicen Maths Live Lesson

Circles 2 • Equation of a Circle • P1 Ex6C • 🚀

Circles 3 • Completing the Square • P1 Ex6C • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 7-14✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 446a: Appreciate that a point on the circumference of a circle centred at the origin satisfies its equation.
• 446b: Plot the graph of a circle centred at the origin.
• 446c: Determine the radius of a circle given its equation.
• 446d: Determine the equation of a circle centred at the origin given its radius.
• 446e: Determine the equation of a circle centred at the origin which is drawn.
• 446f: Determine the equation of a circle centred at the origin, using a point given on the circumference.
• 496a: Determine the equation of a circle, not centred at the origin, given its centre and an integer radius.
• 496b: Determine the equation of a circle, not centred at the origin, which is drawn.
• 496c: Write down the equation of a circle, not centred at the origin, given its centre and radius as a surd.
• 496d: Determine the centre and radius of a circle given in completed square form.
• 496e: Determine the equation of a circle, with given centre and radius/diameter, in expanded form.
• 496f: Plot the graph of a circle not centred at the origin.
• 496g: Appreciate that a point on the circumference of a circle, not centred at the origin, satisfies its equation.
• 496h: Determine the intercepts of circle given its equation.
• 496i: Determine a constant in the equation of a circle given a point on the circumference.
• 496j: Determine the equation of a circle using the endpoints of the diameter.
• 496k: Complete the square to determine the centre and radius of a circle.
• 496l: Determine the valid range of values for a constant in the equation of a circle.
• 496m: Determine the equation of a circle using simple reasoning to establish the centre or radius of the circle.
• 446: Exam Practice: Graphs of circles centred at the origin
• 496: Exam Practice: Coordinate geometry on circles and the equation of a circle with any centre

Other Videos

The Equation of a Circle

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Online Platforms (with Videos & Exercises)

• Khan Academy
  • Conic Sections

Activities

• Mr Etheridge Circle Equation Match (TES Account Required)
• Underground Maths Teddy Bear✦ (similar to Mr Etheridge), Pairs of Circles✦ (Good Extension Problem)
• Ben Sparkes Geogebra Interactive Activity
• The Chalkface
  • Jupiter's Moons: Excel · PowerPoint (if macros are blocked: right-click the file › Properties › tick Unblock)
• Transum
  • Circle Equations✦
• Goal Free Problems
  • A-Level Algebra & Coordinate Geometry set Q11
• Desmos
  • Circle Patterns (write equations of circles to extend a pattern)
  • Equations of Circles (by Jo Hill: write equations of circles from different given information)
• Mathematics Assessment Project
  • Equations of Circles 1✦
• Susan Wall (Active A-level Maths)
  • Circle Geometry: Circles 1 · Circles 2 · Circles 3 (Susan Wall;

Worksheets

• Dr Austin Maths
  • Radius and Diameter Fill in the Blanks (Word | PDF | Answers)
  • Practice Strips (Word | PDF | Answers)
  • Fill In The Blanks (Word | PDF | Answers)
• Jethwa Maths Starter Activities (p5-6)
• Mathematics Assessment Project Equations of Circles 2✦ (p14 Onwards)
• kutasoftware Worksheets
  • Graphing & Properties of Circles
  • Equations of Circles
  • Circles
• MathedUp
  • Circles: Questions · Challenge · Answers
• Maths Genie
  • Equation of a Circle: Questions · Solutions · Lesson · Topic page
• MadAsMaths (T. Madas)
  • Circle Coordinate Geometry Exam Questions Q1, 3–5, 8, 12, 14, 20, 33–34, 38–39, 53, 55–56, 58, 60, 64, 66–67, 69–70, 81, 86, 88, 97, 101, 104, 106, 111–112 (difficulty 2–5 of 5)
  • Curve Sketching Q10 (difficulty 3 of 5)
  • Conic Sections Circle Q3, 13, 18, 19, 23 (difficulty 4–5 of 5)
• SRWhitehouse
  • Equations of circles worksheets✦
• Pret Homework (Jo Morgan)
  • Equation of a Circle✦ (Jo Morgan, docx)

AS Maths – Pure – Intersections of Straight Lines and Circles

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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6D Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (From 9:51 to 13:28)

Bicen Maths Live Lesson

Circles 4 • Intersections with Lines • P1 Ex6D • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

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(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 15-17✦

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 447d: Determine the coordinates of the point where a tangent with given equation intersects a circle centred at the origin. (not yet active)
• 496q: Determine the shortest distance between two circles with given equations.
• 496r: Determine the shortest distance between a line and a circle with given equations. (not yet active)
• 496s: Determine whether circles overlap or touch.
• 496v: Determine whether a straight line cuts or is tangent to a circle.
• 447: Exam Practice: Equation of a tangent to a circle centred at the origin
• 496: Exam Practice: Coordinate geometry on circles and the equation of a circle with any centre

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Worksheets

• martahidegkuti.com Circle Problems 1 (with Notes Beforehand)
• Jethwa Maths Starter Activities (p1-2)
• The Chalk Face Simultaneous Line and Curve
• MadAsMaths (T. Madas)
  • Circle Coordinate Geometry Exam Questions Q2, 6, 10–11, 18, 21–24, 44, 46, 49, 57, 62, 68, 74, 76, 78–79, 84, 87, 94, 108–109 (difficulty 2–5 of 5)
  • Discriminant Exam Questions Standard Q17, 30; Enrichment Q8 (difficulty 4–5 of 5)
  • Conic Sections Circle Q1, 4, 5, 17, 20, 21 (difficulty 3–5 of 5)

AS Maths – Pure – Tangents and Chords to Circles

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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6E Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (From 13:28)

Bicen Maths Live Lesson

Circles 5 • Theorem 1: Tangent meets Radius at Right Angles • P1 Ex6E • 🚀

Circles 6 • Theorem 2: Perpendicular Bisector of Chord Passes through Centre • P1 Ex6E • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

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• Dr Frost Slides 18-24✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 442c: Know that the angle in a semicircle is 90 degrees.
• 442e: Know that the angle between a tangent and the radius at the point of contact is 90.
• 447a: Determine the equation of a tangent to a circle centred at the origin.
• 447b: Determine the x-intercept or y-intercept of a tangent to a circle centred at the origin.
• 447c: Determine the area of a triangle formed by a tangent to a circle centred at the origin.
• 496o: Determine the coordinate of a centre of a circle given its radius and the length of a tangent.
• 496p: Determine the tangent to a circle centred at any point.
• 496t: Solve problems involving lengths of or angles between tangents to a circle not centred at the origin.
• 496u: Determine the shortest distance from a chord of the circle of given length to the centre of the circle.
• 496w: Determine the equation of a tangent to a circle that is parallel to another given tangent.
• 442: Exam Practice: Circle theorems
• 447: Exam Practice: Equation of a tangent to a circle centred at the origin
• 496: Exam Practice: Coordinate geometry on circles and the equation of a circle with any centre

Practice Questions

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Worksheets

• Jethwa Maths Starter Activities (p3-4 & 7-8)
• Solomon C2 Co-Ordinate Geometry C Questions & Solutions
• MadAsMaths (T. Madas)
  • Circle Coordinate Geometry Exam Questions Q7, 9, 13, 15–17, 19, 25–27, 29–32, 35–36, 42–43, 45, 47–48, 51–52, 63, 65, 72–73, 77, 80, 82–83, 89–93, 96, 98–100, 103, 110, 113 (difficulty 2–5 of 5)
  • Geometric Mensuration Q87 (difficulty 4 of 5)
  • Line Coordinate Geometry Exam Questions Enrichment Q17 (difficulty 5 of 5)
  • Conic Sections Circle Q6, 8–11, 16, 27 (difficulty 4–5 of 5)

AS Maths – Pure – Circles and Bisectors

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

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6F Slides (Courtesy of Owen134866)

Dr Frost Video

• Whole Topic Video (From 13:28)

Bicen Maths Live Lesson

Circles 7 • Theorem 3: Right Angled Triangle in a Semi-Circle • P1 Ex6F • 🚀

Circles 8 • Theorem 4: Perpendicular Bisectors of Inscribed Triangle meet at Centre • P1 Ex6F • 🚀

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 25-29✦

Extra Questions

• Dr Frost Questions (Very Short & Limited)

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 496n: Use the coordinates of the endpoint of chord of a circle and the radius to determine the circle’s centre.
• 496x: Determine the equation of a circle given the coordinates of 3 points.
• 496: Exam Practice: Coordinate geometry on circles and the equation of a circle with any centre

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Underground Maths Finding Circles✦
• Math = Love (Sarah Carter)
  • Archaeology Broken Plate Activity
• Goal Free Problems
  • A-Level Algebra & Coordinate Geometry set Q12

Whole Topic Videos

Circles 9 • Exam Questions pt. 1 • P1 Ex6 Mixed • 🚀

Circles 10 • Exam Questions pt. 2 • P1 Ex6 Mixed • 🚀

Bicen Maths Exam Questions

Activities

• @bucksburnmaths Increasingly Difficult Circle Equations
• NRICH (student problems)
  • Coordinate Geometry & Graphs
• Mr Draper Maths Circles Completion Table
  • Circles Completion Table Resource
• @mrsouthernmaths
  • Lines, parabolas and areas task✦
• Andy Lutwyche
  • Erica’s Errors

Worksheets

• martahidegkuti.com Circle Problems 1
• martahidegkuti.com Circle Problems 2✦ (Some Good Extension Qs Here) 
• Mathshelper (Jon Stone)
  • Circles
• Maths Genie
  • Circles: Questions · Solutions · Lesson · Topic page
• Jethwa Maths
  • Year 1 Coordinate Geometry: Worksheets · Unit Test
• Physics & Maths Tutor
  • Circles: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• MME
  • C1 Coordinate Geometry of Curves: Questions | Answers
• Dr Oliver Mathematics
  • Circles: Circles questions✦ (solutions✦)
• StudyWell
  • Equation of a circle exam questions with solutions✦
• MadAsMaths (T. Madas)
  • Circle Coordinate Geometry Exam Questions Q37, 41, 59, 102, 105 (difficulty 3–5 of 5)
  • Conic Sections Circle Q12, 14, 15, 22, 25, 26 (difficulty 5 of 5)
• Dr Austin Maths
  • Line Geometry Revision Practice Grid (Word | PDF | Answers)

External Whole Topic Past Paper Style Questions

• Jethwa Maths Worksheet (p4-7), Topic Test (p5-9)
• Dr Frost Topic Test✦ (Teacher Log in Required)
• Mathsaurus Mixed Exam Board PPQs
• Naiker Maths ✦Exam Qs (2005-2011) & MS
• PMT
  • Summary Straight Lines PPQs
  • AQA C1 Coordinate Geometry PPQs
  • Edexcel Set 1 Co-Ordinate Geometry – Circles PPQs (Old Spec)
  • Edexcel Set 2 Co-Ordinate Geometry – Circles PPQs (Old Spec)
  • OCR MEI C1 Co-Ordinate Geometry – Circles 1 PPQs & MS (Old Spec)
  • OCR MEI C1 Co-Ordinate Geometry – Circles 2 PPQs & MS (Old Spec)
  • OCR MEI C1 Co-Ordinate Geometry – Circles 3 PPQs & MS (Old Spec)
  • OCR MEI C1 Co-Ordinate Geometry – Circles 4 PPQs & MS (Old Spec)
  • Summary Cheat Sheet