Question

Около окружности описана прямоугольная трапеция, боковые стороны которой равны 20 см и 25см. Найдите площадь четырехугольника, вершинами которого являются точки касания окружности со сторонами трапеции

Answer 1

Для описанного 4-угольника суммы противоположных сторон равны)))
20+25 = а+а+15
а = 15 ---меньшее основание трапеции
искомая площадь будет состоять из суммы площадей двух прямоугольных (опирающихся на диаметр окружности) треугольников...
один из этих треугольников будет равнобедренным с катетом, равным
10√2, следовательно, его площадь составит 100
для второго треугольника легко находится высота, проведенная к гипотенузе... она = 8)))
и площадь тогда составит 80
Ответ: 180

Answer 2

Problem Analysis

We are given a circle with an inscribed right trapezoid. The lengths of the trapezoid's lateral sides are 20 cm and 25 cm. We need to find the area of the quadrilateral formed by the points of tangency between the circle and the trapezoid's sides.

Solution

To find the area of the quadrilateral, we can divide it into two triangles and a rectangle. Let's calculate the area of each component and then sum them up.

1. Triangle 1: This triangle is formed by one of the trapezoid's lateral sides, the radius of the circle, and the line segment connecting the point of tangency to the center of the circle. Let's call the radius of the circle 'r'. The area of this triangle can be calculated using the formula: `Area = (1/2) * base * height`. The base of the triangle is the radius 'r', and the height is the length of the lateral side of the trapezoid, which is 20 cm. Therefore, the area of Triangle 1 is `Area1 = (1/2) * r * 20`.

2. Triangle 2: This triangle is formed by the other trapezoid's lateral side, the radius of the circle, and the line segment connecting the point of tangency to the center of the circle. The base and height of this triangle are the same as Triangle 1. Therefore, the area of Triangle 2 is also `Area2 = (1/2) * r * 20`.

3. Rectangle: This rectangle is formed by the line segment connecting the two points of tangency and the line segment connecting the centers of the circle and the trapezoid's lateral sides. The length of the rectangle is the difference between the lengths of the trapezoid's lateral sides, which is 25 cm - 20 cm = 5 cm. The width of the rectangle is the radius of the circle, 'r'. Therefore, the area of the rectangle is `Area3 = length * width = 5 * r`.

To find the total area of the quadrilateral, we sum up the areas of the two triangles and the rectangle: `Total Area = Area1 + Area2 + Area3`.

Let's calculate the area using the given information.

Calculation

Given: - Length of one lateral side of the trapezoid = 20 cm - Length of the other lateral side of the trapezoid = 25 cm

We need to find the area of the quadrilateral formed by the points of tangency between the circle and the trapezoid's sides.

Let's assume the radius of the circle is 'r'.

1. Area of Triangle 1: `Area1 = (1/2) * r * 20` 2. Area of Triangle 2: `Area2 = (1/2) * r * 20` 3. Area of Rectangle: `Area3 = 5 * r` 4. Total Area of the Quadrilateral: `Total Area = Area1 + Area2 + Area3`

Now, let's substitute the values and calculate the area.

Calculation Results

Using the given information, we can calculate the area of the quadrilateral formed by the points of tangency between the circle and the trapezoid's sides.

Let's assume the radius of the circle is 'r'.

1. Area of Triangle 1: `Area1 = (1/2) * r * 20` 2. Area of Triangle 2: `Area2 = (1/2) * r * 20` 3. Area of Rectangle: `Area3 = 5 * r` 4. Total Area of the Quadrilateral: `Total Area = Area1 + Area2 + Area3`

Unfortunately, we don't have enough information to determine the exact value of the radius 'r' or calculate the area of the quadrilateral. The problem statement does not provide any additional information or constraints.

If you have any further information or constraints, please provide them so that we can assist you better.