Question

В равнобедренную трапецию с острым углом 30 вписана окружность радиуса 2 найти площадь трапеции

Answer 1

Пусть а - это длина меньшего осн, b - длина большего основания трапеции. с - длина боковых сторон. h - высота. S=(1/2)*(a+b)*h.

так как окружность вписана в трапецию, то  h=2r=4 и a+b=2c. 

(В четырехугольник можно вписать окружность тогда и только тогда, когда суммы длин его противолежащих сторон равны. Отсюда следует, что если в трапецию вписана окружность, то сумма ее оснований равна сумме боковых сторон.)

S=(1/2)*(2c)*h=c*h по правилу прямоугольного треугольника с(гипотинуза)=h(высота)/sin30=h/(1/2)=2h

S=ch=2*h*h=2*4*4=32.

Answer 2

Calculation of the Trapezoid's Area

To find the area of a trapezoid, we can use the formula:

Area = (a + b) * h / 2

where: - a and b are the lengths of the parallel sides of the trapezoid - h is the height of the trapezoid

In this case, we are given that the trapezoid is isosceles (or "equilateral") with an acute angle of 30 degrees. Additionally, an inscribed circle with a radius of 2 is present.

Let's denote the lengths of the parallel sides as a and b. Since the trapezoid is isosceles, we can assume that a is the longer side and b is the shorter side.

To find the lengths of the sides, we can use the properties of an inscribed circle. The radius of the inscribed circle is given as 2. In an isosceles trapezoid, the diagonals are equal in length and are perpendicular to each other. The diagonals of the trapezoid are also the diameters of the inscribed circle.

Let's denote the length of the diagonal (which is equal to the sum of the parallel sides) as d. Since the trapezoid is isosceles, the diagonal d can be expressed as:

d = a + b

The height of the trapezoid can be calculated using the radius of the inscribed circle. The height is equal to twice the radius of the inscribed circle:

h = 2 * 2 = 4

Now, we can substitute the values of a, b, and h into the formula to calculate the area of the trapezoid.

Area = (a + b) * h / 2 = (d) * 4 / 2 = 2d

To find the value of d, we need to use trigonometry. Since the trapezoid has an acute angle of 30 degrees, we can use the sine function to relate the sides of the triangle formed by the radius of the inscribed circle and the diagonal d.

The formula for the sine of an angle in a right triangle is:

sin(angle) = opposite / hypotenuse

In this case, the opposite side is the radius of the inscribed circle (2) and the hypotenuse is the diagonal d. Rearranging the formula, we get:

d = 2 / sin(30)

Using the value of sin(30) = 0.5, we can calculate d:

d = 2 / 0.5 = 4

Now, we can substitute the value of d into the formula for the area of the trapezoid:

Area = 2d = 2 * 4 = 8

Therefore, the area of the trapezoid is 8 square units.

Answer

The area of the given isosceles trapezoid is 8 square units.