Circle theorems

Author:Valeria Cipolli Kumoto, Jasmine Vincent, PCARTS, Brigitte Gudz, Hillmath, Aidan Pittman, James Monaghan, Brenden, Ishak_Hassan@SG, Jonathan Longstaffe, Caribou Contests

Topic:Circle

Circle Theorem 1 - Angle at the Centre

1) The central angle A and the inscribed angle E are related to the green arc CBD. Move the points C, D, and E to explore the geogebra and then answer the questions below.

a) Write down what you notice about the relationship between the two angles.  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):

Circle Theorem 2: Angles in a semicircle

2) The diameter DB divides the circumference into two equal parts (semicircles). The inscribed angle C is subtended (related) by the red arc DEB. Move point C around the circumference and then move the diameter DB as well. Explore it and answer the questions below.

a) Write down what you notice about the relationship between angle C and arc BD.  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):

Circle Theorem 3 - Inscribed angles subtended by the same arc

3) The Inscribed angles B and D are subtended by the same blue arc AEC. Move the points around the circumference and then answer the questions below.  a) What do you notice?  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):

Circle Theorem 4 - Cyclic Quadrilateral

4) The quadrilateral ABCD is inscribed into the circle. Move the points around the circle and answer the questions below. a) What do you notice?  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):

Circle Theorem 5 - Alternate Segment Theorem

5) Observe the angles between the chords BA and CA and a tangent passing through point A. Move the points A, B, and C around the circumference and answer the questions below.  a) What do you notice?  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):

Circle Theorem 6 - Tangent on a Circle (Tangent and Radius)

6) The blue line is tangent to the circle in the point B and segment AB is the radius of the circle. Move the point B around the circle and answer the questions below. a) What do you notice?  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):

Circle Theorem 7 - Two Tangent Theorem

7) Observe the two red tangents drawn from an external point to the circle passing by points A and B. Move the slider and answer the questions below.  a) What do you notice?  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):

Circle theorem 8 - Radius and Chord

8) Observe the chord CD and the radius AF. Move the points C and D around the circumference.  a) What do you notice?  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):

Extra - Circle Theorem 9 : Intersecting Chord

9) Observe the two chords CD and EF. Move the points C, D, E, and F around the circumference and answer the questions below.  a) What do you notice?  b) Conjecture (idea for what the rule might be):  c) Justification (using rules you already know):