Let \(C\) be a circle with center \(O\). Suppose \(\overline{PR}\) and \(\overline{QS}\) are two diameters of \(C\) which are perpendicular to one another at \(O\) as pictured below:
We draw segments \(PQ\), \(QR\), \(RS\), and \(S\,\) which, collectively, form a square inscribed in the circle as will be shown below:
Commentary
Part (a) of this task could be used for assessment or for instructional purposes. Parts (b) and (c) are mostly for instructional purposes but could be used for assessment. This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context. The problem could be made substantially more demanding by not providing the initial picture and simply prompting students to construct an inscribed square inside a given circle. If used for instructional purposes, the teacher may want to spend some time showing how to construct the two perpendicular diamters of the circle with a straightedge and compass (currently there is no task for this but there will be eventually).
Part 3 of this task is related to a classical computation made by Greek geometers, attempting to estimate the area of a circle by using inscribed polygons with more and more sides. This will be continued in the task ``Inscribing a hexagon in a circle.''