Question

Вершины ломанной ABD лежат на окружностидиаметром которой является отрезок AB длина отрезка BD в два раза меньше длины диаметра окружности вычислите расстояние между точками D и Bесли известно что длина ломанной ABD равна 12 см

Answer 1

Пусть центр окружности точка О, R -радиус окружности.
По свойствам вписанных углов в оружность (угол, к-ый опирается на диаметр окружности равен 90 градусов), значит уголADB=90

По условию АB=2R,  DB=R
Тругольник ADB прямоугольный , значит
 $AD= \sqrt{AB^2-BD^2} = \sqrt{4R^2-R^2} =R \sqrt{3}$

По условию длина ломанной равна 12
$AB+DB=12$

$R \sqrt{3} +R=12$

$DB=R= \frac{12}{1+ \sqrt{3} }$

Answer 2

Problem Analysis

We are given that the vertices of the polyline ABD lie on a circle, with AB as its diameter. The length of segment BD is half the length of the diameter of the circle. We need to calculate the distance between points D and B, given that the length of the polyline ABD is 12 cm.

Solution

To solve this problem, we can use the properties of a circle and the given information.

Let's denote the length of the diameter of the circle as d. We are given that the length of segment BD is half the length of the diameter, so BD = d/2.

We are also given that the length of the polyline ABD is 12 cm. This means that the sum of the lengths of segments AB and BD is 12 cm. Since AB is the diameter of the circle, AB = d.

Therefore, we can write the equation: AB + BD = 12 cm.

Substituting the values, we have: d + d/2 = 12 cm.

To solve this equation, we can multiply both sides by 2 to eliminate the fraction: 2d + d = 24 cm.

Combining like terms, we get: 3d = 24 cm.

Dividing both sides by 3, we find: d = 8 cm.

Now that we know the length of the diameter of the circle, we can calculate the distance between points D and B. Since AB is the diameter, the distance between D and B is equal to half the length of the diameter: 4 cm.

Conclusion

The distance between points D and B is 4 cm, given that the length of the polyline ABD is 12 cm and the vertices of the polyline lie on a circle with AB as its diameter.