16. Exercise Problems VIII

In right triangle $ABC$, right angled at $C, M$ is the mid-point of hypotenuse $AB$. $C$ is joined to $M$ and produced to a point $D$ such that $DM = CM$. Point $D$ is joined to point $B$.

 

 

Show that:

 

(i) $∆ AMC ≅ ∆ BMD$

 

(ii) $∠DBC$ is a right angle.

 

(iii) $∆ DBC ≅ ∆ ACB$

 

(iv) $CM =  \frac{1}{2} AB$