A parallelogram is a quadrilateral whose opposite sides are parallel and equal. The opposite sides being parallel and equal, forms equal angles on the opposite sides. Diagonals of a parallelogram are the segments which connect the opposite corners of the figure.

Where,

p,q are the diagonals

a,b are the parallel sides

\[\LARGE p=\sqrt{a^{2}+b^{2}-2ab\cos (A)}=\sqrt{a^{2}+b^{2}+2ab\cos (B)}\]

\[\LARGE q=\sqrt{a^{2}+b^{2}+2ab\cos (A)}=\sqrt{a^{2}+b^{2}-2ab\cos (B)}\]

\[\LARGE p^{2}+q^{2}=2(a^{2}+b^{2})\]

### Solved Examples

**Question 1: **

Find the diagonal of a parallelogram with sides 3 cm, 5 cm and angle 45 degrees ?

** Solution: **

Given a = 3 cm

b = 5 cm

angle A = 45°

Formula of diagonal is,

q = $\sqrt{a^{2}+b^2-2ab cosA}$

q = $\sqrt{3^{2} + 5^2 – 2\times 3 \times 5 cos 45}$

q = $\sqrt{34 – 30\times 0.707 }$

q = √12.79

=3.576 cm

Diagonal of parallelogram = 3.576 cm.