School level

# From Bisector to Circumcenter

### Investigating Perpendicular Bisector

https://www.projectmaths.ie/geogebra/bisector-line/ Added: 09 Feb 2021 Contributor: PDST Resource type: Simulation

GeoGebra file to explore the properties and relationships within the construction of the perpendicular bisector.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Students use the check boxes to explore the relationships. Drag the points to see if the relationships hold true for any set of values.

Asset type: Online Resource
Language: English
Special Needs:

### Support Questions:

• What is the relationship between C and A,B?
• What could you say about the position of C in relation to A and B?
• Classify the triangles.
• Can you identify any congruent triangles?
• Can you prove that these triangles are congruent?
• Is there another way to prove this?

### Exploring the Circumcenter

https://www.projectmaths.ie/geogebra/geometry-circumcenter/ Added: 09 Feb 2021 Contributor: PDST Resource type: Simulation

Problem-Solving Task using a GeoGebra file to help students to construct the circumcenter.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Present the problem, provide link to the GeoGebra file for students to work on the problem.

Asset type: Online Resource
Language: English
Special Needs:

### Circumcenter Activity

This task follows on from the investigation of the perpendicular bisector at Junior Cycle.
Note: For students to work with this file it would be useful to download or open it in the app once they have thought about the question.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Asset type: Document
Language: English
Special Needs:

### Support Questions:

• Where would you put the fire between two tents?
• How do you know this is the fairest location?
• Use GeoGebra to show the measure of the length of the line segments.
• Is there any other point that is also the same distance from the two tents?
• Repeat for two different tents.

### Synthetic Geometry:

Prior Knowledge: (JC OL)
Definition 27; The perpendicular bisector of a segment [AB] is the line through the midpoint of [AB], perpendicular to AB.
Proposition 10: Each point on the perpendicular bisector of a segment [AB] is equidistant from the ends.
Construction 2: Perpendicular bisector of a segment, using only compass and straight Edge.

(JC HL)
Definition 41: A cyclic quadrilateral is one whose vertices lie on some circle.
Corollary 5: If ABCD is a cyclic quadrilateral, then opposite angles sum to 180◦.

(LC OL)
Theorem 21 (2): The perpendicular bisector of a chord passes through the centre.
Proposition 17: If a circle passes through three non-collinear points A, B, and C, then its centre lies on the perpendicular bisector of each side of the triangle ABC.
Definition 43: The circumcircle of a triangle ABC is the circle that passes through its vertices. Its centre is the circumcentre of the triangle, and its radius is the circumradius.
Construction 16: Circumcentre and circumcircle of a given triangle, using only straight- edge and compass.

Denition 41: A cyclic quadrilateral is one whose vertices lie on some circle.
Corollary 5: If ABCD is a cyclic quadrilateral, then opposite angles sum to 180.

Coordinate Geometry:
Recognise that (x-h)^2 + (y-k)^2 = r^2 represents the relationship between the x and y coordinates of points on a circle with centre (h, k) and radius r.

### Extension Questions

• Under what circumstances would the circumcentre as the position for the fire not be suitable?
• Classify the triangle. Where is the fire located in each case?
• Would this solution work if there were more than 3 tents?
• What is the relationship between the synthetic and coordinate geometry in this task?
• Using algebra justify your solution (equation of a line, point of intersection, distance formula etc.)

### Exploring the properties of the Circumcenter

https://www.geogebra.org/classic/s5s7q8ca Added: 04 Mar 2021 Contributor: PDST Resource type: Simulation

Having engaged with the tents problem students can now investigate the properties of the circumcenter by dragging the points and recording the effect on the location of the circumcenter.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 4. Transformation Geometry, Enlargements

Suggestions for use: Ask students to move the file and consider the following questions:
Under what circumstances would the circumcentre as the position for the fire not be suitable?
Classify the triangle. Where is the fire located in each case?

Asset type: Online Resource
Language: English
Special Needs:

### The Circumcircle (Relationships)

https://www.geogebra.org/classic/ux88byfd Added: 04 Mar 2021 Contributor: PDST Resource type: Simulation

This file allows students to investigate the relationships between the synthetic and coordinate geometry of the circumcircle and circumcenter.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 2. Co-ordinate Geometry

Suggestions for use: Students can drag the points and record the relationship between the algebra and the synthetic geometry of the circumcircle.
Consider questions such as:
What is the relationship between the synthetic and coordinate geometry in this task?
Using algebra justify your solution (equation of a line, point of intersection, distance formula etc.)

Asset type: Online Resource
Language: English
Special Needs:

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