Question

The diagram below shows equilateral triangle ABC sharing a side with square ACDE. The square has side lengths of 4. What is BE? Justify your answer.

Answer 1

So, we know the sides AB = 4, and AE = 4.

The angle EAB is 90+60 = 150.

The law of the cosines helps:

BE^2 = AE^2 + AB^2 - 2*AB*AE*cos(150) = 4^2 + 4^2 - 2*4*4*(-sqrt(3)/2)

BE^2 = 16+16+16*sqrt(3) = 16(2+sqrt(3)),

BE = 4*sqrt( 2 + sqrt(3 ) ) Nice number

BE ~ 7.7274 ... ~ 7.73


In fact, if you take a ruler, one can see that the ratio between AB and BE is ~ 1.95 which s the same number.

Answer: BE = 4*sqrt( 2 + sqrt(3 ) ) ~ 7.73