Question

A quadrilateral is obtained by joining the mid-points of a rectangle. Further, a rectangle is obtained by joining the mid points of quadrilateral obtained above. Again another quadrilateral is obtained by joining the mid points of the second rectangle. This process is repeated for an infinite number of times. Find the ratio of the sum of areas of all the rectangles to the sum of the areas of all the quadrilaterals.

• A. 2:1
• B. 3:1
• C. 4:1
• D. 2:3

Answer 1

The correct option is A

2:1

The quadrilateral formed by joining the mid-points of a rectangle is a rhombus.
Area of rhombus = $\frac{1}{2}$ * area of rectangle

Area of inner rectangle = $\frac{1}{2}$ * (area of rhombus)

Let the area of the biggest rectangle be 1 square unit.
Therefore, Sum of area of all the rectangles obtained = 1 + $\frac{1}{4}$ + $\frac{1}{16}$ + ... + $\infty$ = $\frac{1}{(1-\frac{1}{4})}$ = $\frac{4}{3}$ square units

Sum of area of all the rhombuses = $\frac{1}{2}$ + $\frac{1}{8}$ + $\frac{1}{32}$ + ........$\infty$ = $\frac{\frac{1}{2}}{1-\frac{1}{4}}$ = $\frac{2}{3}$
$\therefore$ Required ratio = $\frac{\frac{4}{3}}{\frac{2}{3}}$ = 2:1