Show that the diagonals of a square are equal and bisect each other at right angles.
Solution:
Let’s suppose that we are given a square ABCD with diagonals AC and BD.
In
and
, we have
(Alternate Interior angles)
AB=CD (Sides of square are equal)
(Alternate Interior angles)
Therefore, by ASA congruence rule,
It means that we have AO=CO and BO=DO (1)
(Corresponding parts of congruent triangles are equal)
In
and
AO=AO (Common)
AB=AD (Sides of square are equal)
OB=OD (Proved above)
Therefore, by SSS congruence rule,
(Corresponding parts of congruent triangles are equal) (2)
But, we also have
(Linear pair) (3)
Using equation (2) in equation (3), we get
(4)
Similarly,
(5)
From (1), (4) and (5), we can say that diagonals of a square are equal and bisect each other at right angles.
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