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
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