684/VB03. problem / Math - Solved Math Problems

MR-684 / 4. problem

The sides of the rectangle are 3cm and 1cm. All four bisectors of its angles are constructed to their mutual intersections. Calculate the area of the resulting quadrilateral whose vertices are the intersecting points of the bisectors.

Source: Vene T. Bogoslavov: Collection of Solved Problems #3 - 4..)

Geometry → Two-dimensional geometric shapes → Quadrilaterals

$x^{2} + x^{2} = 1^{2}$

$2 x^{2} = 1$

$x^{2} = \frac{1}{2}$

$x = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$

$x = \frac{1}{\sqrt{2}} | \cdot \frac{\sqrt{2}}{\sqrt{2}}$

$x = \frac{\sqrt{2}}{2}$

$A = {(2 x)}^{2}$

$A = 4 x^{2}$

$A = 4 {(\frac{\sqrt{2}}{2})}^{2}$

$A = 4 \cdot \frac{{(\sqrt{2})}^{2}}{2^{2}}$

$A = 4 \cdot \frac{2}{4}$

$A = 2$