Download Circle Theorems[ ] Theorem 1a: 1. Open geogebra 2. Make a circle

Circle Theorems
Theorem 1a:
1. Open geogebra
2. Make a circle with centre A and point B
(a) do this by using the “circle with centre through point” button
3. Make a point C on the circle
(a) use the “new point” button
(b) click on the circle where you want the point
(c) this point should appear as a lighter blue
4. Make a segment between points B and C
(a) use the drop-down menu from “line between two points”
(b) use the “segment between two points” button
(c) click on point B
(d) click on point C
5. Make the midpoint D of the line BC
(a) use the drop-down menu from “new point”
(b) use the “midpoint or centre” button
(c) click on point B
(d) click on point C
6. Make a line perpendicular to BC which passes through D
(a) use the “perpendicular line” button
(b) click on line a (i.e. line BC) - NOT point B or C
(c) click on point D
7. What other point does line b pass through?
8. Move around points A, B, C. Does line b always pass through that point?
(a) use the “move” button
9. Sketch this drawing in the space below.
This activity created by Ms. Dunwoody at Vedanta Academy.
Theorem 1b:
Click on file → new or file → new window
Create a circle with centre A and point B
Make a point C on the circle
Make a segment between points B and C
Make a line that passes through A and is perpendicular to BC
Create a point D at the intersection of BC and this perpendicular line
(a) use the drop-down menu from “new point”
(b) use “intersect two objects” button
(c) click on line BC
(d) click on the perpendicular line
7. measure the distance BD
(a) use the drop-down menu from “angle”
(b) use “distance or length” button
(c) click on point B
(d) click on point D
8. measure the distance CD
9. What do you notice about BD and CD?
10. Move the points A, B and C. Is that always true?
11. Make a general conclusion:
1.
2.
3.
4.
5.
6.
If
and
then
12. Sketch the drawing in the space below.
Theorem 1c:
1.Create a new file
2.Create a circle
3.Create a chord (a segment between two points on the circle)
4.Create a point at the midpoint of the chord
5.Create a line or segment between the midpoint of the chord and the centre of the circle
6.Measure the angle between the chord and the line through the centre
7.What do you notice about that angle?
8.Make a general conclusion.
9.Sketch the drawing in the space below.
Theorem 2:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Open geogebra
Draw a circle
Create two more points (C and D) on the circle
Create segments (not lines) between the following pairs of points:
(a) B and D
(b) C and D
Measure the angle at D
(a) Measure the small angle; if clicking BDC doesn’t work, try CDB
Create one more point (E) on the circle
(a) Create the point between points B and D or between points C and D
Create line segments between the following pairs of points:
(a) B and E
(b) C and E
Measure the (small) angle at E
What do you notice?
10.
Move the points around (especially points D and E). Is this always true?
11.
Sketch the drawing and describe this rule.
Theorem 3:
1. Create a circle
2. Draw two more points (C and D) on the circle
3. Connect the following pairs of points with segments:
(a) B and D
(b) C and D
4. Measure the (small) angle at D
5. Connect the following pairs of points with segments:
(a) B and A
(b) C and A
6. Measure the (small) angle at A
7. What do you notice?
8. Move the points around. Is this always true?
9. Sketch the drawing and explain the general rule.
Theorem 4:
1. Draw a circle
2. Draw a point C somewhere outside the circle
3. Create tangent lines from point C to the circle
(a) Use the drop-down menu from “perpendicular line”
(b) Use the “tangents” button
(c) Click on point C
(d) Click anywhere on the circle (not on point B)
4. Create a new point (D) at a place where one of the tangent lines touches the circle.
5. Create a segment between D and A (the centre of the circle)
6. Measure the angle at D. What is it?
7. Move some of the points (especially point C). What do you notice?
8. Draw the picture (in your picture, you only need to draw one tangent line, not both of
them) and describe the theorem.
Theorem 5:
9. Don't open a new file.
10.
Create a new point (E) at the place where the other tangent line touches the circle.
11.
Measure the distance between points D and C
12.
Measure the distance between points E and C
13.
What do you notice?
14.
Move point C. What do you notice?
15.
Draw the picture and describe the theorem.
Theorem 6:
1.
2.
3.
4.
5.
6.
7.
8.
Create a new file. (or delete the radius, the angle, the distances and point E)
Create a circle.
Create a point C outside the circle.
Create tangent lines to the circle.
Create a point D at the intersect of one of the tangent lines to the circle.
Create a point E somewhere on the circle.
Create a segment between points D and E
Measure the acute (less than 90 degrees) angle between the chord and the tangent line
(a) This is sometimes easiest if you first create a point on the tangent line
9. Create an inscribed angle subtended by the chord
(a) Remember that inscribed means all three points on the angle touch the circle
(b) Remember that if it's subtended by the chord, two of the points have to be that chord
(c) Start by creating another point (F) anywhere on the circle
10.
Measure the acute angle at F
11.
What do you notice?
12.
As you move point F, you don't expect that angle to change. Which theorem from
yesterday tells you this?
13.
Move point C or E. What do you notice?
14. Draw the picture (again, you only need to draw one tangent line) and describe the
theorem.
Theorem 7:
1. Create a circle.
2. Create an inscribed quadrilateral
(a) Inscribed means all the corners have to touch the circle
(b) Quadrilateral means it has four corners
3. Measure one of the interior (i.e. inside the circle) angles of the quadrilateral
4. Measure the opposite interior angle of the quadrilateral
5. Find the sum of these angles
(a) Click on the “input” button at the bottom of the window (if it's not a button, click on
the space to the right of “input”)
(b) Click on one of your angles (if this doesn't cause the letter alpha to appear to the
right of the “input” button, click on alpha from the drop-down menu at the bottom of
the window)
(c) Type “+”
(d) Click on the other angle
(e) Push the “enter” key on your keyboard
(f) A new angle (gamma) should appear on the list to the left of the window
6. What is angle gamma?
7. What does this tell you about your two angles?
8. Move around various points in your drawing. Under what conditions does gamma stay
at 180○?
9. Sketch your drawing and describe the theorem.
Circle Theorems Key
Theorem 1a:
7. the centre of the circle
8. yes
9.
suggested discussion:
statement of result:
If a line is perpendicular to a chord
and the line passes through the middle of the chord (the line bisects the chord)
then the line passes through the centre of the circle
Theorem 1b:
9. They have the same length.
10. Yes
11. If a line is perpendicular to a chord
and it passes through the centre of the circle
then is cut the chord exactly in half (it bisects the chord)
12.
Theorem 1c:
7. The angle is 90◌.
8. If a line cuts a chord exactly in half (a line bisects a chord)
and the line passes through the centre of the circle
then the line is perpendicular to the chord (the line is at 90◌ to the chord)
9.
suggested discussion:
name the theorem
Theorem 2:
9. The two angles are the same.
10. Yes – as long as the two angles are not on opposite sides of the circle
11. Inscribed angles standing on the same two points and on the same side of the circle are
the same.
or Inscribed angles subtended by (“standing on”) the same arc are equal.
suggested discussion:
name the theorem (the pizza theorem, the pie theorem, the mountain theorem)
Theorem 3:
7. The angle at the centre of the circle is two times the angle at the edge of the circle (the
central angle is twice the inscribed angle)
8. Yes – as long as the two angles are not on opposite sides of the circle
9. A central angle standing on the same two points and on the same side of the circle as an
inscribed angle is twice the size of the inscribed angle.
or A central angle is twice an inscribed angle subtended by (“standing on”) the same arc.
suggested discussion:
name the theorem (Star Trek theorem, taco theorem)
discuss the consequence of this … an inscribed angle subtended by a diameter will be 90○.
Students can check this if desired, or the teacher can create a file already set up to check this
new theorem on a shared drive accessible to students.
Name the second theorem (cheese theorem)
Theorem 4:
6.
90○
7. it stays the same
8. The angle between a tangent and a radius at the point of tangency is 90○.
Suggested discussion:
Name the theorem (pacman faceplant theorem – you can see this if you draw the tangent
vertical and imagine pacman with his mouth closed)
Theorem 5:
13. The two lengths are equal.
14. The lengths remain equal.
15. The lengths of two tangents from the same point to their points of tangency are equal.
Suggested discussion:
Name the theorem (ice cream cone theorem)
Theorem 6:
11. The angles are equal
12. The pizza / pie / mountain / inscribed angles theorem
13. The angles remain equal
14. The angle between a tangent and a chord at the point of tangency is the same as an
inscribed angle subtended by (“standing on”) that same chord.
Suggested discussion:
Name the theorem (the see-saw theorem)
Theorem 7:
6.
180○
7. Angles alpha and beta add to 180 (or the angles are supplementary)
8. As long as alpha and beta are on opposite sides of the circle.
9. The opposite angles in a cyclic quadrilateral add up to 180.
Suggested discussion:
Name the theorem (the origami theorem, the baseball theorem)