Question

In Fig. the line segment joining the mid-points M and N of opposite sides AB and DC of quadrilateral ABCD is perpendicular to both these sides. Prove that the other sides of the quadrilateral are equal. [4 MARKS]

Answer 1

Concept : 1 Mark
Process : 2 Marks
Proof : 1 Mark


Join M and D and also M and C

In $\Delta$ CMN and $\Delta$ DMN,
DN = NC [N is the mid-point of CD]
$\angle$DNM = $\angle$CNM = ${90}^{\circ }$
And, MN = MN [Common]

$\Delta$CMN $\cong$$\Delta$DMN [SAS congruence criterion]
$\Rightarrow$ CM = MD ------(i) [C.P.C.T.]
and $\angle$CMN = $\angle$DMN ------(ii) [C.P.C.T.]

Now, $\angle$AMN=$\angle$BMN [Each equal to ${90}^{\circ }$ (Given)]
$\Rightarrow$ $\angle$AMD = $\angle$CMB [From (ii)]

Thus, in $\Delta$s AMD and BMC, we have
AM = MB [M is the mid-point of AB]
$\angle$AMD = $\angle$CMB [Proved above]
And, DM=MC [Proved above]

$\Delta$AMD $\cong$ $\Delta$ CMB [ SAS congruence criterion]
$\Rightarrow$ AD = BC.