Problem:
Amy painted a dart board over a square clock face using the "hour positions" as boundaries. If t is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and q is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then tq​=
Answer Choices:
A. 23​−2
B. 23​
C. 25​+1​
D. 3​
E. 2
Solution:
Let O be the center of the clock, and label the triangle from 12 o'clock to 1 o'clock AOB, the quadrilateral from 1 o'clock to 2 o'clock OBCD, and the 3 o'clock position E, as indicated in the figure. Then △AOB≅△EOD. Let AB=1. Since ∠AOB=30∘, it follows that OA=3​, [OACE]=3 and [AOB]=3​/2. Hence tq​=
Note that ∠AOB=∠BOD=30∘. Let AO=1. Therefore AB=3​1​ and [ABO]=23​1​. Draw OC. Compute the area of △OBC using BC=1−3​1​ as the base and AO=1 as the altitude. Then
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