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)
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.)
It is given that and (1)
We have (2) (Corresponding angles)
From (1) and (2), we get
In and , we have
(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