Question

# Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Solution

## Given: $$\Box ABCD$$ is a quadrilateral.diag $$AC =$$ diag $$BD$$, intersecting at $$E$$.$$AC$$ and $$BD$$ are perpendicular bisectors of each other$$\therefore \angle E=90^o$$To prove: $$\Box ABCD$$ is a square.Solution: In $$\triangle ABE$$ and $$\triangle ADE$$.$$BE = DE$$              ....given$$AE = AE$$              ...common side$$\angle AEB \cong \angle AED$$     ....each $$90^o$$$$\therefore \triangle ABE \cong \triangle ADE$$     ...SAS test of congruence$$\therefore AB = AD$$                  ...c.s.c.t.      ....(1)Similarly, we can prove $$\triangle ABE \cong \triangle CBE$$$$\therefore AB = CB$$    ....c.s.c.t.        ....(2)And, $$\triangle ADE \cong \triangle CDE$$$$\therefore AD = CD$$    ....c.s.c.t.        ....(3)$$\therefore$$ From (1), (3) and (4),$$AB = CB = CD = AD$$     ............(4)Thus, in quad.ABCD$$AB = CB = CD = AD$$  and   $$AC=BD$$ [given]$$\therefore \Box ABCD$$ is a square        ....By definitionMathematicsRS AgarwalStandard IX

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