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25th March

Geometry - Circle, Tangents, Chords

Learning Outcomes

Students working at OL should be able to:

  • perform constructions 16-21 (see Geometry for Post-primary School Mathematics)
  • investigate theorems 7, 8, 11, 12, 13, 16, 17, 18, 20, 21 and corollary 6 (see Geometry for Post-primary School Mathematics) and use them to solve problems
  • recognise that (x - h)² + (y - k)² = r² represents the relationship between the x and y co-ordinates of points on a circle with centre (h, k) and radius r
Prior Learning
Making Connections
Equation of a Circle
Geogebra Files
Circle Properties
SEC Exam Questions

Bisecting a Line Segment - Instructional Video

https://www.youtube.com/watch?v=tFgUn-l1_Cs Added: 10 May 2018 Contributor: PDST Resource type: Demonstration / Audio / Video

This video is designed to support teachers to understand the construction for bisecting a line segment. It also considers how to deal with this construction in terms of teaching and learning in the classroom, its connections going forward and links to the Literacy & Numeracy Strategy

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JC Mathematics
SC Mathematics
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JC
Mathematics

Strand: 3. Geometry & Trigonometry

Strand unit: 3. Geometrical Proof

Suggestions for use: used to demonstrate how to construct the perpendicular bisector of a line segment.

Key Skills: Being Creative Communicating Managing Information and Thinking Working with Others

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: used to demonstrate how to construct the perpendicular bisector of a line segment.

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Construction 19

https://www.scoilnet.ie/pdst/maths/...etry-trigonometry/constructions/19/ Added: 25 Mar 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

Tangent to a given circle at a given point on it.

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Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Geogebra file to explore the construction of a tangent to a circle.

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Theorem 20

https://www.scoilnet.ie/pdst/maths/...ry-trigonometry/theorems/theorem20/ Added: 25 Mar 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

(i) Each tangent is perpendicular to the radius that goes to the point of contact.
(ii) If P lies on the circle S, and a line l is perpendicular to the radius to P, then l is a tangent to S.

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SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Geogebra file to explore theorem 20:
(i) Each tangent is perpendicular to the radius that goes to the point of contact.
(ii) If P lies on the circle S, and a line l is perpendicular to the radius to P, then l is a tangent to S.

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Theorem 21

https://www.scoilnet.ie/pdst/maths/...ry-trigonometry/theorems/theorem21/ Added: 25 Mar 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

(i) The perpendicular from the centre to a chord bisects the chord.
(ii) The perpendicular bisector of a chord passes through the centre.

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SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Geogebra file to explore theorem 21:
(i) The perpendicular from the centre to a chord bisects the chord.
(ii) The perpendicular bisector of a chord passes through the centre.

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Corollary 6

https://www.youtube.com/watch?v=z-YxfG42P2M Added: 25 Mar 2020 Contributor: PDST Resource type: Audio / Video

If two circles intersect at one point only, then the two centres and the point of contact are collinear.

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SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Video exploring tangents to two circles.

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Constructions - Leaving Certificate

https://www.scoilnet.ie/pdst/maths/...eometry-trigonometry/constructions/ Added: 09 Nov 2019 Contributor: PDST Resource type: Guide/notes / Simulation

Interactive Geogebra files and students worksheets demonstrating the Leaving Cert Constructions 16 - 22.

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Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

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The equation of a circle

https://www.scoilnet.ieuploads/resources/32041/31783.pptx Added: 25 Mar 2020 Contributor: PDST Resource type: Presentation

Using pythagoras' theorem to find the equation of a circle

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Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 2. Co-ordinate Geometry

Suggestions for use: This helpful presentation uses a constructivist approach to lead students to discover the equation of a circle using pythagoras' theorem. The presentation contains links to geogebra files, and also makes explicit the connections between geometry, coordinate geometry and trigonometry.

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Intersection of a line and a circle

https://www.geogebra.org/m/fe17ayul Added: 25 Mar 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

Geogebra file to explore the intersection of a line and a circle using coordinate geometry

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SC Mathematics
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SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 2. Co-ordinate Geometry

Suggestions for use: Explore the intersection between a line (ax+by+c=0) and a circle.
Adjust the value of the Radius to give the desired Circle.
Then adjust the value of a, b and c to give the desired Line ax+by+c=0.
Check the point(s) of intersection M and N at the bottom of the screen.

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Tangent to a Circle

https://www.geogebra.org/m/guIKaQYm Added: 25 Mar 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

In a circle, the angle between the radius and tangent of the circle is always 90 degrees.

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SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Explore the properties of a circle and tangent to the circle. Use this geogebra file to show that a tangent always meets the radius of a circle at a 90 degree angle.

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Equation of a circle with centre (0, 0)

https://www.geogebra.org/classic/jr2steag Added: 25 Mar 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

Use Pythagoras' theorem to discover the equation of a circle with centre (0, 0)

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SC Mathematics
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Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 2. Co-ordinate Geometry

Suggestions for use: This geogebra file can be used as a demonstration or an exploratory tool to show that pythagoras' theorem can be used to discover the equation of a circle.

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Equation of a circle with centre (h, k)

https://www.geogebra.org/classic/ukufacyx Added: 25 Mar 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

Use Pythagoras' theorem to discover the equation of a circle when it's centre is not at (0, 0).

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SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 2. Co-ordinate Geometry

Suggestions for use: This geogebra file can be used as a demonstration or an exploratory tool to show that pythagoras' theorem can be used to discover the equation of a circle.

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Circle Properties - Part 3

https://www.youtube.com/watch?v=jCBMXHQBvEA Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / Video

Video investigates the properties of a circle by examining the following points:
E) Tangents
F) Perpendicular to a chord.

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SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

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SEC 2018 OL paper 2 Q4 - Circle & Tangent Question

https://www.scoilnet.ieuploads/resources/32055/31797.pdf Added: 25 Mar 2020 Contributor: PDST Resource type: Exam Papers

SEC exam paper question to assess knowledge of the circle and tangent

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Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 2. Co-ordinate Geometry

Suggestions for use: SEC past exam question

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SEC 2019 OL paper 2 Q4 - Circle & Tangent Question

https://www.scoilnet.ieuploads/resources/32057/31799.pdf Added: 25 Mar 2020 Contributor: PDST Resource type: Exam Papers

SEC past exam question using a square of 4 tangents of a circle to find the equation of that circle.

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Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 2. Co-ordinate Geometry

Suggestions for use: SEC past exam question to assess understand of the circle and tangents to the circle.

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