GeoGebra file to explore the properties and relationships within the construction of the perpendicular bisector.View full description
Problem-Solving Task using a GeoGebra file to help students to construct the circumcenter.View full description
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.
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.
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◦.
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.
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.
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.View full description
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?
This file allows students to investigate the relationships between the synthetic and coordinate geometry of the circumcircle and circumcenter.View full description
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.)
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