ABCD is a rectangle in which diagonal AC bisects
as well as
. Show that:
(i) ABCD is a square (ii) diagonal BD bisects
as well as
.
Solution (i)
Let’s suppose that we have rectangle ABCD in which diagonal AC bisects
as well as
.
We have AB=CD and BC=AD (Opposite sides of rectangle are equal) (1)
In
and
(Given)
AC=AC (Common)
(Given)
Therefore, by ASA congruence rule,
(Corresponding parts of congruent triangles are equal) (2)
From (1) and (2), we can say that ABCD is a rectangle having all the sides equal. It means that ABCD is a square.
Solution (ii)
In solution (i), we have showed that ABCD is a square.
Now in
and
BC=BA (Sides of square are equal)
BD=BD (Common)
CD=AD (Sides of square are equal)
Therefore, by SSS congruence rule,
(Corresponding parts of congruent triangles are equal) (3)
And,
(Corresponding parts of congruent triangles are equal) (4)
From (3) and (4), we can say that BD bisects
as well as
.
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