Question

Доведіть що коли радіус кола, вписаного в правильний восьмикутник, дорівнює R, то радіус кола описаного навколо нього, дорівнює R/cos 22,5

Answer 1

Ответ:

Обычно радиус вписанной окружности обозначают r, а радиус описанной R. Решение записал исходя из этих обозначений.

Объяснение:

Answer 2

Relationship between the Radius of an Inscribed Circle and the Radius of a Circumscribed Circle in a Regular Octagon

To prove that when the radius of a circle inscribed in a regular octagon is equal to R, the radius of the circle circumscribed around the octagon is equal to R/cos 22.5°, we can use the properties of a regular octagon and trigonometry.

A regular octagon is a polygon with eight equal sides and eight equal angles. In a regular octagon, the center of the circle inscribed within it coincides with the center of the octagon. Similarly, the center of the circle circumscribed around the octagon also coincides with the center of the octagon.

Let's consider a regular octagon with a circle inscribed within it. The radius of the inscribed circle is R. We want to find the radius of the circle circumscribed around the octagon, which we'll denote as R'.

To find the relationship between R and R', we can use trigonometry. In a regular octagon, each interior angle measures 135°. If we draw a line from the center of the octagon to one of its vertices, we create a right triangle with one angle measuring 67.5° (half of 135°). This angle is formed by the radius of the inscribed circle and the radius of the circumscribed circle.

Using trigonometric ratios, we can determine that the cosine of 67.5° is equal to the ratio of the adjacent side (R') to the hypotenuse (R). Therefore, we have:

cos 67.5° = R' / R

To find the value of cos 67.5°, we can use the half-angle identity for cosine:

cos 67.5° = cos (45° + 22.5°) = cos 45° cos 22.5° - sin 45° sin 22.5°

Using the trigonometric values of cos 45° and sin 45° (which are equal), we can simplify the equation:

cos 67.5° = (1/√2) cos 22.5° - (1/√2) sin 22.5°

Now, we substitute this value back into the previous equation:

(1/√2) cos 22.5° - (1/√2) sin 22.5° = R' / R

To find the value of R', we multiply both sides of the equation by R:

R' = R * [(1/√2) cos 22.5° - (1/√2) sin 22.5°]

Therefore, the radius of the circle circumscribed around the regular octagon is equal to R multiplied by [(1/√2) cos 22.5° - (1/√2) sin 22.5°].

Please note that the trigonometric values used in this explanation are approximations and may vary slightly depending on the level of precision required.