Question

Help me solve it please

Answer 1

Answer:

$\fbox{\begin{minipage}{5.8 em}x =5.97 cm\end{minipage}}$

Step-by-step explanation:

(1) As definition, the area ($A$) of a circle is calculated by:

$A = \pi$ x $r^{2}$ with $r$ is radius of circle.

(2) The relation between radius ($r$) and diameter ($d_{1}$) of a circle is:

$d_{1}$ = 2 x $r$

(3) Applying the Pythagorean theorem for a square with sides of length $x$,

the relation between side $x$ and diagonal $d_{2}$ is:

$x^{2} +x^{2} = d^{2} \\or\\2x^{2} = d^{2}$

=> The diagonal ($d_{2}$) of that square is calculated by:

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

(4) The square fits exactly inside a circle, with each of its vertices being on the circumference of the circle.

Applying the theorem of inscribed angle in a circle, the measure of arc BD is twice as the measure of inscribed angle BAD.

In other words, the measure of arc AC = 2 x 90 = 180 deg, or half of circle.

It means that the diagonal BD is coincide with a diameter of circle.

=> $d_{1} = d_{2}$

or $2$ x $r$ = $x\sqrt{2}$

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

Substitute $r$ back into original formula used for calculating the area of circle:

=> A = $\pi$ x $(x\frac{\sqrt{2}}{2})^{2}$ = $\pi$ x $\frac{x^{2}}{2}$

=> $x^{2} = A$ x $\frac{2}{\pi }$

=> $x = \sqrt{A}$ x $\sqrt{\frac{2}{\pi } }$ = $\sqrt{56}$ x $\sqrt{\frac{2}{\pi } }$ = 5.97 cm

Hope this helps!

:)