In trapezoid $ABCD$ with $AB \parallel CD$ and $AB=3, CD=7,$ and $AD=BC,$ define $M$ to be the midpoint of side $BC.$ The circle with diameter $DM$ is tangent to line $AB.$ What is the length of the altitude from $AB$ to $CD.?

I started by drawing a roughly accurate diagram on geogebra.

It seems that the point of tangency of the circle with diameter $DM$ to $AB$ is a trisection point, but I'm not sure. I also drew $MG$ the midline of $ABCD.$ Then if $h$ is the height from $A$ to $MG$ the area of $ABCD$ is $4h+6h=10h.$ Then it suffices to find the area of $ABCD.$ But I'm not sure how to do this. Furthermore I'm not sure how to relate the circle with diameter $DM$ to the problem, all I know is that if $F$ is the center then $\angle FEA=90$ and $\angle DEM = 90.$ May I have some help? Thanks in advance.

In trapezoid $ABCD$ with $AB \parallel CD$ and $AB=3, CD=7,$ and $AD=BC,$ define $M$ to be the midpoint of side $BC.$ The circle with diameter $DM$ is tangent to line $AB.$ What is the length of the altitude from $AB$ to $CD$?

I started by drawing a roughly accurate diagram on geogebra.

It seems that the point of tangency of the circle with diameter $DM$ to $AB$ is a trisection point, but I'm not sure. I also drew $MG$ the midline of $ABCD.$ Then if $h$ is the height from $A$ to $MG$ the area of $ABCD$ is $4h+6h=10h.$ Then it suffices to find the area of $ABCD.$ But I'm not sure how to do this. Furthermore I'm not sure how to relate the circle with diameter $DM$ to the problem, all I know is that if $F$ is the center then $\angle FEA=90$ and $\angle DEM = 90.$ May I have some help? Thanks in advance.