+797 \overline{NM} is the median of trapezoid ABCD. If [ABMN] = 8 and [BNCD] = 12, what is $\frac{CD}{AB}?$
+129933 BNCD is a triangle...h = the height of the trapezoid
[ BNCD ] = (1/2) h ( CD)
12 = (1/2) h (CD)
24 = h * CD
CD = 24/h
ABMN is a trapezoid
Height of ABMN = (1/2) height of the trapezoid = (1/2)h
[ ABMN] = (1/2) [ (1/2) ( h) ( AB + MN) ]
8 = (1/2) [ (1/2)(h) ( AB + (AB + CD) / 2)]
8 = (1/4)h * [ (3/2)AB + (1/2)CD ]
32 = h * ( (3/2) AB + (1/2)CD)
32 = (1/2)h * [ 3AB + CD ]
64 = h * [3AB + CD]
64 = h * [ 3AB + 24/h]
64 = h * 3AB + 24
40 = h * 3AB
40/3 = h * 3AB
40 / [ 3h ] = AB
CD / AB = [ 24/h ] [ 40 / (3h) ] = 72 / 40 = 9 / 5
