Consider only triangle DAB.
AD^2 + AB^2 = (3x)^2, where x=DP = PQ = QB
AD^2 + AB^2 = 9x^2
Any combination of the sum for AB^2 and AD^2 must be a factor of 9x^2. Let's plug in x=1
(A) \(\sqrt{2}\) to 1 ---> 2+1=3, a factor of 9
(B) \(\sqrt{3}\) to 1 ---> 3+1=4, not a factor of 9
(C) \(\sqrt{3}\) to \(\sqrt{2}\) ---> 3+2=5, not a factor of 9
(D) 2 to 1 ---> 4+1=5, not a factor of 9
(E) 2 to \(\sqrt{3}\) ---> 4+3=7, not a factor of 9
Hence, answer is
AGeminiHeat wrote:
In the figure above, ABCD is a rectangle, and each of AP and CQ is perpendicular to BD. If DP = PQ = QB, what is the ratio of AB to AD?
(A) \(\sqrt{2}\) to 1
(B) \(\sqrt{3}\) to 1
(C) \(\sqrt{3}\) to \(\sqrt{2}\)
(D) 2 to 1
(E) 2 to \(\sqrt{3}\)