Question

Периметр правильного шестиугольника вписанного в окружностьравен 48 найдите сторону квадрата вписанного в ту же окружность

Answer 1

Длина сторона правильного шестиугольника вписанного в окружность, в точности равна радиусу этой окружности. Следовательно,  радиус окружности R =P/6 = 48/6 = 8. Диагонали квадрата, вписанного в эту окружность, будут являться диаметрами, и будут пересекаться в центре окружности под прямым углом. Тогда прямоугольный треугольник, образованный стороной (а) вписанного квадрата  и двумя радиусами (R) окружности будет равнобедренный. По теореме Пифагора a²= R² +R² = 2R²= 2*8² = 2*64 = 128. Тогда сторона квадрата а = √128 = 8√2  

Answer 2

Perimeter of a Regular Hexagon Inscribed in a Circle

To find the side length of a square inscribed in the same circle, we first need to determine the perimeter of the regular hexagon inscribed in the circle.

The perimeter of a regular hexagon is given by the formula: Perimeter = 6 * Side Length.

Given that the perimeter of the hexagon is 48, we can set up the equation:

48 = 6 * Side Length

To find the side length, we can rearrange the equation:

Side Length = 48 / 6 = 8

Therefore, the side length of the regular hexagon inscribed in the circle is 8.

Side Length of a Square Inscribed in the Same Circle

To find the side length of a square inscribed in the same circle, we can use the relationship between the side length of the square and the diameter of the circle.

In a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. The diagonal of a square can be found using the formula: Diagonal = Side Length * √2.

Since the diagonal of the square is equal to the diameter of the circle, we can set up the equation:

Diagonal = Diameter = Side Length * √2

Given that the side length of the square is the same as the side length of the hexagon (8), we can substitute the values:

8 * √2 = Diameter

To find the side length of the square, we can rearrange the equation:

Side Length = Diameter / √2

To calculate the diameter, we can use the formula for the circumference of a circle: Circumference = π * Diameter.

Given that the perimeter of the hexagon is 48, we can calculate the circumference of the circle using the formula:

Circumference = 48

Using the circumference formula, we can solve for the diameter:

48 = π * Diameter

Diameter = 48 / π

Now we can substitute the value of the diameter into the equation for the side length of the square:

Side Length = (48 / π) / √2

Calculating this value gives us the side length of the square inscribed in the circle.

Please note that the exact value of the side length will depend on the approximation used for the value of π.