Problem 20 Complex Numbers

Problem 20: Complex Numbers

AIME 12 2005A #14

Problem: Consider the points A(0, 12), B(10, 9), C(8, 0), and D(−4, 7). There is a unique square S such that each of the four points is on a different side of S. Let K be the area of S. Find the remainder when 10K is divided by 1000. Note: General knowledge of operations with complex coordinates is assumed in this document. Also, =(a + bi) is used to denote the imaginary part of a + bi. Similarly, <(a + bi) is used to denote the real part of a + bi. Solution: We can use complex numbers to solve this problem, but first we must convert the given coordinates to complex coordinates. We get A = 12i, B = 10 + 9i, C = 8, D = −4 + 7i. We notice that the unique square in the problem does not have sides parallel to the axes. We want to rotate the square so that its sides are parallel to the axes. We do that by multiplying each point by an unknown angle θ. Our new points are now A0 = (12i)cisθ, B 0 = (10+9i)cisθ, C 0 = (8)cisθ, and D0 = (−4 + 7i)cisθ. Now that our square has axes parallel to the axes, we notice that the difference in height of A’ and C’ is the difference in their imaginary parts. Also, this is the same length as the difference between the real parts of B 0 and D0 , and it is also the side length of the square, which we will call x. We can now write the equation x = =(A0 − C 0 ) = <(B 0 − D0 ), and start to simplify: =(12icisθ − 8cisθ) = <((10 + 9i)cisθ − (−4 + 7i)cisθ) 12 cos θ − 8 sin θ = 10 cos θ − 9 sin θ + 4 cos θ + 7 sin θ 2 cos θ = −6 sin θ sin θ 1 =− cos θ 3 Using the identity sin2 θ + cos2 θ = 1, we get sin θ =

−1 √ 10

and cos θ =

Plugging these values back into our equation, we find x =

√44 . 10

√3 . 10

We need to find the area of

442 10 .

the square, which will be The problem asks for 10 times this value, which is 442 = 1936, and thus our answer is 936 .

Solution was written by Sean Soni and compiled from Art of Problem Solving Forums.