Question
Show that the diagonals of a rhombus bisect each other at right angles.
Solution
Let ABCD be the rhombus. Take A as origin.
Let
be the position vectors of B and D respectively so that
Now,
[
BC is equal and parallel to AD]
Position vector of mid-point of diagonal AC is
Also position vector of mid-point of diagonal BD is
Let
be the position vectors of B and D respectively so thatNow,
[
BC is equal and parallel to AD]Position vector of mid-point of diagonal AC is
Also position vector of mid-point of diagonal BD is
∴ position vector of mid-point of diagonal AC is same as position vector of mid-point of diagonals BD.
∴ diagonals AC and BD bisect each other.
Also, 
[
AD = AB as all sides of rhombus are equal]
diagonals AC and BD are perpendicular to each other.
Hence the result.
