**ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that**

**(i) D is the mid-point of AC (ii) (iii) **

** **

**Solution (i) **

It is given that M is the mid-point of side AB and .

Therefore, **by converse of mid-point theorem**, D is the mid-point of side AC.

**(The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.)**

**Solution (ii)**

It is given that and ** (1)**

We have **(2) (Corresponding angles)**

From **(1)** and **(2)**, we get

.

**Solution (iii)**

In and , we have

**(Common) **

** (Each equal to )**

** (D is the mid-point of AC as proved above in solution (i))**

Therefore,** by SAS congruence rule**, we have

**(Corresponding parts of congruent triangles are equal) (3)**

It is given that M is the mid-point of side AB which means that ** (4)**

From **(3)** and **(4)**, we can say that