Problem:
Vertex E of equilateral △ABE is in the interior of unit square ABCD. Let R be the region consisting of all points inside ABCD and outside △ABE whose distance from AD is between 31​ and 32​. What is the area of R ?
Answer Choices:
A. 7212−53​​
B. 3612−53​​
C. 183​​
D. 93−3​​
E. 123​​ Solution:
Draw a line parallel to AD through point E, intersecting AB at F and intersecting CD at G. Triangle AEF is a 30−60−90∘ triangle with hypotenuse AE=1, so EF=23​​. Region R consists of two congruent trapezoids of height 61​, shorter base EG=1−23​​, and longer base the weighted average
{OR}
Place ABCD in a coordinate plane with B=(0,0),A=(1,0), and C=(0,1). Then the equation of the line BE is y=3​x, so E=(21​,23​​), and the point of R closest to B is (31​,33​​). Thus the region R consists of two congruent trapezoids with height 61​ and bases 1−23​​ and 1−33​​. Then proceed as in the first solution.
