## A Quarter Circle Investigation, Explanation & Generalization

Investigation
A quarter of a circle with center at point O is given, and AO and BO are perpendicular radii.
The point C is any point on the small arc BA of the circle.
The segment AD is perpendicular to chord BC extended (ADCD), as shown below.
1) Drag point C along arc BA. What do you observe? Can you make any conjectures?
2) Click on the 'Show Measurement' button. What do you notice about the lengths of DC and DA?
3) Can you determine the value of ∠BCA from the given information?
4) Check your calculation in 3) by clicking on the 'Show Angle Measurement' button.
5) If necessary, continue dragging to check your observations above.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } Quarter Circle Investigation

Challenge
1) Can you explain why (prove that) ∠BCA = 135°?
2) Can you explain, using a circle geometry theorem, why ∠BCA remains constant (invariant) when dragged along arc BA?
3) Can you explain why (prove that) DC = DA?

Generalize
4) Explore what happens in general when ∠AOB and ∠BDA are supplementary angles. Click on the 'Link to Generalize' button to move to a new dynamic sketch.
5) What do you notice about the lengths of DC and DA in this case? Check your observation not only by dragging C on arc BA, but also change ∠AOB between (0°, 180°) by dragging either A or B.
6) Can you explain why (prove that) your observation in 5) is true?
7) Can you figure out how point D was located (constructed) so that ∠BDA is supplementary to ∠AOB? Click on the 'Show Construction' button to check your answer.
8) Check your observations further by more dragging.
9) When would △CDA be equilateral? Why?

Variation
10) Investigate what happens when (instead of the construction in 4)-8) above), a perpendicular is dropped from A to the line BC to meet it at D. Click on the 'Link to Variation' button to move to a new dynamic sketch.
11) What do you notice about the angles of the formed right triangle CDA as C is dragged along on arc BA? What can you therefore conclude about all the right triangles CDA (for a fixed ∠AOB)? Specifically, what geometric relationship exists between all the right triangles CDA (when ∠AOB is kept constant)?
12) For what value(s) of ∠AOB will the right triangle CDA be a 30°, 60°, 90° triangle?
13) Does the same geometric relationship between all right triangles CDA observed in 11) also hold for all the triangles CDA in the preceding cases?

Further Variation
14) Investigate what happens when ∠CAD has a fixed value when C is dragged along on arc BA. Click on the 'Link to Further Variation' button to move to a new dynamic sketch.
15) What do you notice about the angles of the formed triangle CDA as C is dragged along on arc BA? What can you therefore conclude about all these triangles CDA (for a fixed ∠AOB)? Specifically, what geometric relationship exists between all triangles CDA (when ∠AOB is kept constant)?
16) When would △CDA be equilateral? Why?
17) What is the locus (path) of D as C is dragged along on arc BA? Click on the 'Animate Point C' button to check.
18) Can you logically explain (prove) your observation in 17)? Does it also apply to all the other earlier cases?

Published Paper
Stupel, M. & De Villiers, M. (2023). A Quarter Circle Investigation, Explanation & Generalization. International Journal for Technology in Mathematics Education, Vol 30, No 2 (June), pp. 109-111.

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