Question

Дана прямоугольная трапеция, основания которой равны 2 см и 6 см. В нее вписан круг. Найти площадь круга.

Answer 1

Известно, в четырехугольник можно вписать окружность только, если суммы противоположных  сторон равны.
у нас сумма длин оснований 2+6=8, значит сумма боковых сторон также 8.

Высота прямоугольной трапеции равна 2R. Разность оснований 6-2=4. Выражаем длину наклонной боковой стороны L по т. Пифагора. Она равна:

√4^2+(2R)^2=√16+4R^2

8-L=2R ⇒ L=8-2R⇒L^2=16+4R^2=64-32R+4R^2⇒
64-32R=16⇒32R=48⇒R=48/32=1.5   

площадь круга πR^2 =2.25 π

Answer 2

Problem Analysis

We are given a trapezoid with bases measuring 2 cm and 6 cm. A circle is inscribed within this trapezoid. We need to find the area of the circle.

Solution

To find the area of the circle, we need to know the radius of the circle. Since the circle is inscribed within the trapezoid, the radius will be equal to the perpendicular distance between the center of the circle and any of the parallel bases of the trapezoid.

Let's denote the radius of the circle as r.

To find the radius, we can use the fact that the radius, the perpendicular distance, and the height of the trapezoid form a right triangle. We can use the Pythagorean theorem to find the radius.

The height of the trapezoid can be found by subtracting the length of the smaller base from the length of the larger base.

Let's denote the height of the trapezoid as h.

Once we have the radius, we can use the formula for the area of a circle to find the area.

Calculation

Given: - Length of the smaller base = 2 cm - Length of the larger base = 6 cm

To find the height of the trapezoid: - h = Length of the larger base - Length of the smaller base - h = 6 cm - 2 cm - h = 4 cm

To find the radius of the circle: - Use the Pythagorean theorem: r^2 = (h/2)^2 + (6-2)^2 - r^2 = (4/2)^2 + 4^2 - r^2 = 2^2 + 4^2 - r^2 = 4 + 16 - r^2 = 20 - r = sqrt(20) - r ≈ 4.47 cm

To find the area of the circle: - Use the formula for the area of a circle: A = πr^2 - A = π(4.47)^2 - A ≈ 62.83 cm^2

Answer

The area of the circle inscribed within the given trapezoid is approximately 62.83 cm^2.