cos(x)=AD/3, sin(x)=BD/3, cos(y)=CD/7, sin(y)=BD/7.
cos2(x)+sin2(x) = AD2/9 + BD2/9 = 1.
cos2(y)+sin2(y) = CD2/49 + BD2/49 = 1.
BD2 = 9 - AD2 = 49 - CD2.
Remember that AD + CD = 9, or CD = 9 - AD.
9 - AD2 = 49 - (9-AD)2.
9 = 49 - 81 + 18AD.
AD = 41/18.
(41/18)2 + BD2 = 9.
BD = (9-(41/18)2)1/2 = 1.95236
Others have pointed out an alternate solution using Heron's Formula, which states the area of a triange is sqr(s*(s-a)*(s-b)*(s-c)), where s=(a+b+c)/2, and a, b, and c are the lengths of the three sides.
In this case a, b, and c are 3, 7, and 9.
s=(3+7+9)/2=9.5
area=sqr(9.5*(9.5-3)*(9.5-7)*(9.5-9)) = sqr(1235/16) = 8.785641695
The area must also equal (1/2)*9*h, where h is the height.
So sqr(1235/16) = (1/2)*9*h
h=sqr(1235/16)*(2/9) = 1.95236
The Wizard of Odds