Question

Равнобедренная трапеция с боковой стороной ,равной 18 дм,и углом при основании 60 градусов описана около круга.Найти длины оснований трапеции и её площадь.

Answer 1

Там "стандартное решение" этой задачи - в комментарии. Если это можно назвать задачей. Мне тут в голову пришло, как её устно решить на 10 секунд, поэтому публикую, чтобы не пропало :)
Вообще-то такую трапецию (ну, с точностью до размеров, то есть подобную) можно построить только одним способом - взять правильный треугольник, вписать в него окружность, и провести к этой окружности касательную "сверху".
Эта касательная отсечет меньший правильный треугольник, размеры которого в 3 раза меньше исходного (а почему?)
То есть, если "боковая сторона трапеции" 18, то основания 27 и 9 (ну, 27 - 9 = 18, 27/3 = 9, считайте, что я подобрал числа), а площадь (1 - (1/3)^2)*27^2*√3/4, или, если так проще - то (по площади, как ВОСЕМЬ отсеченных треугольников...)  8*9^2*√3/4 = 162√3; :)

Answer 2

Problem Analysis

We are given a trapezoid with one of its side lengths equal to 18 dm and an angle of 60 degrees at the base. The trapezoid is circumscribed around a circle. We need to find the lengths of the bases of the trapezoid and its area.

Solution

To solve this problem, we can use the properties of a circumscribed trapezoid and trigonometry.

Let's denote the lengths of the bases of the trapezoid as a and b. We can use the following properties:

1. The diagonals of a circumscribed trapezoid are equal in length. 2. The diagonals of a trapezoid divide each other proportionally.

Using these properties, we can set up the following equation:

`(a + b) / 2 = 18` Simplifying the equation, we get:

`a + b = 36`

Since the angle at the base of the trapezoid is 60 degrees, we can use trigonometry to find the lengths of the bases. Let's consider the right triangle formed by the height of the trapezoid, the radius of the circumscribed circle, and half the difference of the bases.

Using trigonometry, we have:

`sin(60) = (a - b) / (2r)`

Simplifying the equation, we get:

`(a - b) = 2r * sin(60)`

Since the trapezoid is circumscribed around a circle, the radius of the circle is equal to half the difference of the bases. Therefore, we can rewrite the equation as:

`(a - b) = 2 * (a + b) * sin(60) / 2`

Simplifying the equation, we get:

`(a - b) = (a + b) * sin(60)`

Expanding the equation, we get:

`a - b = a * sin(60) + b * sin(60)`

Rearranging the equation, we get:

`a - a * sin(60) = b * sin(60) + b`

Factoring out a and b, we get:

`a * (1 - sin(60)) = b * (1 + sin(60))`

Dividing both sides by (1 - sin(60)), we get:

`a = b * (1 + sin(60)) / (1 - sin(60))`

Using the value of sin(60) = sqrt(3) / 2, we can substitute it into the equation:

`a = b * (1 + sqrt(3) / 2) / (1 - sqrt(3) / 2)`

Simplifying the equation, we get:

`a = b * (2 + sqrt(3)) / (2 - sqrt(3))`

Now, we can substitute the value of a in terms of b into the equation a + b = 36 to solve for b:

`(b * (2 + sqrt(3)) / (2 - sqrt(3))) + b = 36`

Simplifying the equation, we get:

`b * (2 + sqrt(3)) + (2 - sqrt(3)) * b = 36 * (2 - sqrt(3))`

Expanding the equation, we get:

`2b + b * sqrt(3) + 2b - b * sqrt(3) = 72 - 36 * sqrt(3)`

Simplifying the equation, we get:

`4b = 72 - 36 * sqrt(3)`

Dividing both sides by 4, we get:

`b = (72 - 36 * sqrt(3)) / 4`

Simplifying the equation, we get:

`b = 18 - 9 * sqrt(3)`

Now, we can substitute the value of b into the equation a + b = 36 to solve for a:

`a + (18 - 9 * sqrt(3)) = 36`

Simplifying the equation, we get:

`a = 36 - (18 - 9 * sqrt(3))`

Simplifying the equation, we get:

`a = 18 + 9 * sqrt(3)`

Now, we have the lengths of the bases of the trapezoid. To find the area of the trapezoid, we can use the formula:

`Area = (a + b) * h / 2`

where h is the height of the trapezoid.

Since the trapezoid is circumscribed around a circle, the height of the trapezoid is equal to the radius of the circle. Therefore, we can substitute the value of the radius of the circle into the equation:

`Area = (a + b) * r / 2`

Substituting the values of a, b, and r, we get:

`Area = (18 + 9 * sqrt(3) + 18 - 9 * sqrt(3)) * r / 2`

Simplifying the equation, we get:

`Area = 36 * r / 2`

Simplifying the equation, we get:

`Area = 18 * r`

Therefore, the area of the trapezoid is 18r.

Answer

The lengths of the bases of the trapezoid are 18 + 9 * sqrt(3) dm and 18 - 9 * sqrt(3) dm. The area of the trapezoid is 18r.

Note: The value of r is not given in the problem statement, so we cannot calculate the exact area without knowing the value of r.