Question

If a rectangle and a parallelogram are equal in area and have the same base and are situated on the same side then the quotient: $\frac{\text{Perimeter of rectangle}}{\text{Perimeter of parallelogram}}$ is

• A. Equal to $1$
• B. Greater than $1$
• C. Less than $1$
• D. Indeterminate

Answer 1

The correct option is C Less than $1$

In the figure $ABCD$ is a rectangle and $AB{C}^{{}^{\prime }}{D}^{{}^{\prime }}$ is a parallelogram.
$AB$ is the common base.
$AB=a$, $AD=b$, $A{D}^{{}^{\prime }}=b^{{}^{\prime }}$
Area of rectangle $ABCD=ab$
and area of parallelogram $AB{C}^{{}^{\prime }}{D}^{{}^{\prime }}=ab$
Here, we see that $b^{{}^{\prime }}>b$
Thus, $\frac{\text{Perimeter of rectangle}}{\text{Perimeter of}\parallel gm}=\frac{2(a+b)}{2(a+b^{{}^{\prime }})}$
$\Rightarrow b^{{}^{\prime }}>b$
Thus, $a+b^{{}^{\prime }}>a+b$
Thus, $\frac{2(a+b)}{2(a+b^{{}^{\prime }})}<1$