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Question

Prove that the rhombus with equal diagonals is a square.


Solution


$$ABCD$$ is a rhombus, in which diagonals AC and BD are equal.
We know that diagonals of rhombus bisect each other.
As $$AC=BD$$
$$\therefore AO=BO=CO=DO$$
In $$\triangle AOB,$$
$$\Rightarrow  AO=OB$$ and $$\angle AOD=90°$$
$$\therefore \angle OAB=\angle OBA=\dfrac{90^o}{2}=45°$$
Similarly in $$\triangle AOD,$$
$$\Rightarrow  \angle OAD=\angle ODA=45°$$
$$\therefore \angle A=\angle OAB+\angle OAD=45^o+45^o=90°$$
Similarly,
$$\Rightarrow \angle B=\angle C=\angle D=90°$$
here, $$AB=BC=CD=DA$$
$$\therefore$$ Quadrilateral ABCD is a square.

Mathematics

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