- Define the dimensions of a square
- Inscribe a rhombus whose vertices touch the midpoint of each side of the square, and indicate the size of the rhombus
- Indicate the area of the square
- Indicate the area of the rhombus
See development and solution
Development:
- A square is defined with its side $$l=6$$ cm.
-
Note that the axes of the inscribed rhombus ($$D$$ and $$d$$) measure the same as the side of the square $$l$$. $$$D=6 \ \mbox{cm}$$$ $$$d=6 \ \mbox{cm}$$$
-
The area of the square is: $$$A=(6 \ \mbox{cm})^2=36 \ \mbox{cm}^2$$$
- The area of the rhombus is: $$$A_{rhombus}= \dfrac{D\cdot d}{2}=18 \ \mbox{cm}^2 = \dfrac{A_{square}}{2} $$$
See that the inscribed rhombus is also a square of side $$\sqrt{18}=3\sqrt{2}.$$
Thus, the side of the rhombus could also have been calculated (with the Pythagorean theorem) and then squared to obtain the area.
Solution:
- $$l=6$$ cm
- $$D=6$$ cm, $$d=6$$ cm
- $$A=36 \ \mbox{cm}^2$$
- $$A_{rhombus}= 18 \ \mbox{cm}^2$$