Problem: A B C D A B C D M M N N B C B C C D C D sin β‘ ΞΈ = sin ΞΈ =
Answer Choices:
A. 5 5 5 5 β β
B. 3 5 5 3 β
C. 10 5 5 1 0 β β
D. 4 5 5 4 β
E. none of these answers
Solution:
We may suppose that the sides of the square have length 2 2 B M = N D = 1 B M = N D = 1
sin β‘ ΞΈ = sin β‘ ( Ο 2 β 2 Ξ± ) = cos β‘ 2 Ξ± = 2 2 Ξ± β 1 = 2 ( 2 5 ) 2 β 1 = 3 5 sin ΞΈ β = sin ( 2 Ο β β 2 Ξ± ) = cos 2 Ξ± = 2 cos 2 Ξ± β 1 = 2 ( 5 β 2 β ) 2 β 1 = 5 3 β β
Alternately, one may express Area β³ A M N β³ A M N sin β‘ ΞΈ sin ΞΈ β³ A M N β³ A M N sin β‘ ΞΈ sin ΞΈ
