The midpoints of the adjacent sides of a rhombus will form a rectangle. State whether the given statement is true or false.
The midpoints of the adjacent sides of a rhombus will form a rectangle. State whether the given statement is true or false.
The correct option is A True
Let ABCD be the rhombus and let PQRS be the quadrilateral formed by joining the midpoint of adjacent sides of the rhombus.
We know that diagonals AC and BD of the rhombus are perpendicular to each other.
In ∆ABC, by mid point theorem, we can say that PQ || AC and PQ = 12 AC
In ∆ADC, by mid point theorem, we can say that SR || AC and SR = 12 AC
Combining these two, we can say that PQ = SR and PQ || SR.
Since one pair of opposite sides of PQRS are equal and parallel, PQRS is a parallelogram.
Now consider ∆ABD. By mid point theorem, PS ||BD.
Since, BD is perpendicular to Ac and PS is perpendicular to AC.
And since AC || PQ, PS is perpendicular to PQ.
Thus, the parallelogram PQRS has one right angle in it.
Hence, it is a rectangle.
True
Let ABCD be the rhombus and let PQRS be the quadrilateral formed by joining the midpoint of adjacent sides of the rhombus.
We know that diagonals AC and BD of the rhombus are perpendicular to each other.
In ∆ABC, by mid point theorem, we can say that PQ || AC and PQ = 12 AC
In ∆ADC, by mid point theorem, we can say that SR || AC and SR = 12 AC
Combining these two, we can say that PQ = SR and PQ || SR.
Since one pair of opposite sides of PQRS are equal and parallel, PQRS is a parallelogram.
Now consider ∆ABD. By mid point theorem, PS ||BD.
Since, BD is perpendicular to Ac and PS is perpendicular to AC.
And since AC || PQ, PS is perpendicular to PQ.
Thus, the parallelogram PQRS has one right angle in it.
Hence, it is a rectangle.
