Question

Refer to the pdf file for the referenced shapes.
(a) How many different (whether congruent or not) triangles can we inscribe in the circle in this way?
(b) How many different quadrilaterals can we incribe in the circle by using three or more of the marked vertices?
(c) How many different polygons of three or more sides can we inscribe in the given circle by using 3 or more of the marked vertices?

Answer 1

refer to this file.

Answer 2

(a) Here I got C(8,3) as there are
8 points to use (N) with lines going through
3 points (r) to form a triangle.
(b) Here I got C(8,4). The 4 for r comes
from the 4 points needed to make the
quadrilateral.
(c) And finally, for the last one,
I ended up using summation notation to get
\[\sum_{3}^{i}\left(\begin{matrix}8 \\ j\end{matrix}\right)\] where j is 3 and i is the ith number of vertices.