**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 (*AD* ⊥ *CD*), 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.

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.

**Related Links**

Matric Exam Geometry Problem - 1949

Cyclic Hexagon Alternate Angles Sum Theorem

A generalization of the Cyclic Quadrilateral Angle Sum theorem

Angle Divider Theorem for a Cyclic Quadrilateral

Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi

Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons

Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral

An extension of the IMO 2014 Problem 4

A Forgotten Similarity Theorem?

Similar Parallelograms: A Generalization of a Golden Rectangle property

A similarity theorem by Wares

A generalization of a theorem for 'equilic' quadrilaterals

A variation of Miquel's theorem and its generalization

Pirate Treasure Hunt and a Generalization

A Fundamental Theorem of Similarity

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Release: 2020Q3, semantic Version: 4.8.0, Build Number: 1070, Build Stamp: 8126303e6615/20200924134412

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Created 29 January 2023 by Michael de Villiers, using *WebSketchpad*; updated 17 Jan 2024.