Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Solution:
Let’s suppose that we are given a quadrilateral ABCD. Diagonals of quadrilateral ABCD bisect each other at right angles.
In
and
(Given)
(Each given equal to
)
(Given)
Therefore, by SAS congruence rule,
It means that we have AB=CD (Corresponding parts of congruent triangles are equal)
(Corresponding parts of congruent triangles are equal)
because
and
are alternate interior angles.
Quadrilateral is a parallelogram if one pair of opposite sides is equal and parallel. It means that quadrilateral ABCD is a parallelogram. (1)
and
(2)
In order to prove that it is a rhombus, we just need to prove that all the sides of parallelogram ABCD are equal.
Now in
and
(Common)
(It is given that diagonals bisect each other at
)
(Given)
Therefore, by SAS congruence rule,
It means that we have AB=AD (Corresponding parts of congruent triangles are equal) (3)
From (1), (2) and (3), we can say that ABCD is a parallelogram having all the sides equal. It means that ABCD is a rhombus.
Leave a Reply