Question

Найдите радиус вписанной окружности и периметр правильного 4-х угольника если радиус описанной окружности равен 16 корень из двух дм

Answer 1

Данная фигура является квадратом по определению.
Радиус описанной окр. квадрата - половина диагонали.
Т.е. диагональ равна $2R=2*16 \sqrt{2} =32 \sqrt{2}$
Обозначим сторону квадрата как $a$, тогда по т. Пифагора:
$a^2+a^2=(32 \sqrt{2} )^2$
$a \sqrt{2} =32 \sqrt{2}$
$a=32$
Радиус впис. окр квадрата равен половине стороны, т.е. 16

Answer 2

I have searched the web for your question and found some relevant results. Here is a summary of what I learned:

- The question is about finding the radius of the inscribed circle and the perimeter of a regular quadrilateral if the radius of the circumscribed circle is equal to 16 times the square root of two decimeters. - A regular quadrilateral is a square (see the picture). AC is the diagonal of the square, equal to the diameter of the circumscribed circle. The diameter of the circumscribed circle is equal to 32 times the square root of two. Then, by the Pythagorean theorem, the side of the square is equal to 32 centimeters. The side of the square is the diameter of the inscribed circle. The radius of the inscribed circle is equal to 32 divided by 2, which is 16 centimeters. The perimeter of the square is equal to 32 times 4, which is 128 centimeters. - The answer is: the radius of the inscribed circle is 16 centimeters, the perimeter of the regular quadrilateral is 128 centimeters.