Question

An ABCD rectangle has a width of 12 cm and a length of 16 cm. The middle points of the edges of this rectangle are combined to create another rectangle inside the ABCD rectangle.
Then the middle points of the new quadrilateral are combined and another quadrilateral is drawn. In this way, by joining the midpoints of the previous square,
rectangles continue to be created. What is the sum of the circles of all quadrilaterals?

sorry i use translate.I bad at english

Answer 1

This problem is based on the Varignon Parallelogram . The theory states that if midpoints of any quadrilteral are joined, the quadrilateral so formed is always a parallelogram whose area is always half the area of the original quadrilateral.

So we can directly use the result, that each successive quadriletral area is half the previous quadrilateral.

16 12

Area of original rectangle = 16 ×12 = 192 sq. cm

Area of next quadrilateral = 192 × 1/2

Area of next quadrilateral = 192 × 1/4

Area of next quadrilateral = 192 × 1/8

......... and so on

Total area = 192 [1 + 1/2 + 1/4 + 1/8 + .... ∞)

The value in racket is an infinite GP, and its sum is 2.

Hence, total area = 192 × 2 = 384 sq. cm