Question

A point O is taken inside an equilateral four sided figure ABCD such that its distances from the angular points D and B are equal. Show that AO and OC are in one and the same straight line. [3 MARKS]

Answer 1

Process : 2 Marks
Proof : 1 Mark

Given: A point O inside an equilateral quadrilateral four sided figure ABCD such that BO = OD

To prove : AO and OC are in one and the same straight line.

Proof: In $\Delta$s AOD and AOB, we have
AD = AB [Given]
AO = AO [Common side]
OD = OB [Given]
SO,
$\Delta$ AOD $\cong$ $\Delta$ AOB. [SSS criterion of congruence]
$\Rightarrow$ $\angle$1 = $\angle$2.... (i) [C.P.C.T.C]

Similarly, $\Delta$ DOC $\cong$ $\Delta$BOC
$\Rightarrow$ $\angle$3 = $\angle$4....(ii) [C.P.C.T.C]


But $\angle$1 + $\angle$2+$\angle$3+ $\angle$4 = ${360}^{\circ }$
$\Rightarrow 2\angle$2+2$\angle$3=${360}^{\circ }$ [Using (i) and (ii)]


$\Rightarrow$ $\angle$2+$\angle$3 = ${180}^{\circ }$
$\Rightarrow$ $\angle$2+$\angle$3 = ${180}^{\circ }$


$\Rightarrow$ $\angle$2 and $\angle$3 form a linear pair
$\Rightarrow$ AO and OC are in the same straight line
$\Rightarrow$ AC is a straight line.