Question

Выручайте товарищи! Найти длину описанной около прямоугольника окружности, если одна сторона 7см, и она образует с диагональю угол в 30 градусов.

Answer 1

Обозначим одну из сторон прямоугольника = 7, другую - х. Так как угол между стороной и диагональю равен 30°, то катет, лежащий против этого угла, равен половине гипотенузы. Тогда по теореме Пифагора: 7²+х²=(2х)²; 3х²=49; х²=49/3; х=7/√3. Ну а радиус описанной вокруг прямоугольника равен половине диагонали, т.е. R=7/(2√3).

Answer 2

Calculation of the Length of the Circumscribed Circle

To find the length of the circumscribed circle around a rectangle, we need to know the length of one side of the rectangle and the angle it forms with the diagonal.

Given: - One side of the rectangle is 7 cm. - The angle between this side and the diagonal is 30 degrees.

To calculate the length of the circumscribed circle, we can use the properties of a right-angled triangle formed by the side, the diagonal, and the radius of the circumscribed circle.

Let's denote: - The length of the side of the rectangle as a (a = 7 cm). - The angle between the side and the diagonal as θ (θ = 30 degrees).

The diagonal of the rectangle can be calculated using the Pythagorean theorem: diagonal = sqrt(a^2 + a^2) = sqrt(2a^2) = sqrt(2) * a.

In this case, the diagonal of the rectangle is: diagonal = sqrt(2) * 7 cm.

The radius of the circumscribed circle is half the length of the diagonal: radius = diagonal / 2 = (sqrt(2) * 7 cm) / 2.

Finally, the length of the circumscribed circle can be calculated using the formula: circumference = 2 * π * radius.

Substituting the value of the radius, we get: circumference = 2 * π * (sqrt(2) * 7 cm) / 2.

Now, let's calculate the length of the circumscribed circle.

Calculation:

Using the given values, we can calculate the length of the circumscribed circle as follows:

a = 7 cm (length of one side of the rectangle)

θ = 30 degrees (angle between the side and the diagonal)

diagonal = sqrt(2) * a = sqrt(2) * 7 cm

radius = diagonal / 2 = (sqrt(2) * 7 cm) / 2

circumference = 2 * π * radius = 2 * π * (sqrt(2) * 7 cm) / 2

Simplifying the expression:

circumference = π * sqrt(2) * 7 cm

Therefore, the length of the circumscribed circle around the rectangle is π * sqrt(2) * 7 cm.

Please note that the above calculation assumes that the rectangle is a right-angled rectangle. If the rectangle is not right-angled, the calculation may differ.

I hope this helps! Let me know if you have any further questions.