12. Constructing a parallelogram with two diagonals and one included angle

Construct a parallelogram \(PQRS\) with \(PR\) \(=\) \(10\) \(cm\), \(QS\) \(=\) \(8\) \(cm\) and \(\angle POQ\) \(=\) \(120^{\circ}\). Also, find its area.

 

Construction:

 

Step 1: Draw a line segment \(PR\) \(=\) \(10\) \(cm\).

 

Step 2: Mark the midpoint of the line segment \(PR\) as \(O\).

 

Step 3: Draw a line \(XY\) through \(O\) such that \(\angle POQ\) \(=\) \(120^{\circ}\).

 

Step 4: With \(O\) as centre, draw two arcs each of radii \(4\) \(cm\) on either side of \(PR\) intersecting \(XY\) at \(Q\) and \(S\).

 

Step 5: Join \(PQ\), \(QR\), \(RS\) and \(SP\).

 

Step 6: \(PQRS\) is the required parallelogram. The measure of \(SA\) gives the height of the parallelogram \(PQRS\).