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Figure A shows one of the Gothic window designs used in Wells Cathedral in England. (The cathedral was constructed in the mid-thirteenth century.) Figure \( B \) indicates how the design is formed. We start with the equilateral \( \triangle A B C \) and construct the equilateral arch \( A B C \). Next, the midpoints of the three sides of \( \triangle A B C \) are joined to create four smaller equilateral triangles. The two equilateral triangles \( A D F \) and \( F E C \) are then used to construct the two smaller equilateral arches shown in Figure B. And finally, the equilateral tria ngle \( D B E \) is used to construct the equilateral curved triangle within the top half of the figure.
(a) Let \( s \) denote the length of a side of the equilateral triangle \( A B C \). Express the area of each of the equilateral arches \( A B C \) and \( A D F \) in terms of s. Also, find the ratio of the area of arch \( A D F \) to arch \( A B C \). \( \operatorname{area}_{A B C} \) \( \operatorname{area}_{A D F} \)
ratio
(b) Express the area of the equilateral curved triangle DBE in terms of s. (c) Express the area of the curved figure DEF in terms of s. (By "the curved figure DEF" we mean the region bounded by the circular arcs \( D E, E F \), and \( F D \).) (d) Express the area of the curved figure BEC in terms of s.