we know that
AB : AC = 3 : 7 and BD : CD = 9 : 2
so
AB/AC=3/7
AC=AB+BC
AB/(AB+BC)=3/7-----> equation 1
BD/CD=9/2-----> equation 2
CD=100 cm-----> substitute in the equation 2
BD/100=9/2-----> 2*BD=100*9-----> divide by 2 both sides----> BD=900/2
BD=450 cm
BD=BC+CD----à BC=BD-CD---à BC=450-100--à BC=350 cm
Equation 1
AB/(AB+BC)=3/7---à AB/(AB+350)=3/7
7*AB=3*(AB+350)
7*AB=3*AB+1050
4*AB=1050
AB=1050/4
AB=262.5 cm
the answer is
the length of AB is 262.5 cm