Question

Хорды МN и MK окружности равны по 18 см, а угол КМN равен 120 градусов Найдите диаметр этой окружности

Answer 1

 Соединив точки М и Н с центром окружности О, получим равнобедренный треугольник, две стороны которого образованы радиусами ОН и ОМ, а третья - хордой НМ. Углы при основании НМ равны 60 градусов, поскольку данный между хордами угол раве 120 градусов, а хорды равны. Отсюда третий угол тоже равен 60 градусов. Мы получили равносторонний треугольник с радиусом, равным хорде и равным 18 см. Диаметр, соответственно равен 36 см.

Answer 2

Problem Analysis

We are given that the chords MN and MK of a circle are both 18 cm long, and the angle KMN is 120 degrees. We need to find the diameter of this circle.

Solution

To find the diameter of the circle, we can use the relationship between the length of a chord and the angle it subtends at the center of the circle. The formula for this relationship is:

Length of chord = 2 * radius * sin(angle/2)

In this case, we are given the length of the chord MN as 18 cm and the angle KMN as 120 degrees. Let's calculate the diameter of the circle using this information.

Calculation

Given: Length of chord MN = 18 cm Angle KMN = 120 degrees

Using the formula mentioned above, we can calculate the diameter of the circle:

Length of chord MN = 2 * radius * sin(angle/2)

Substituting the given values:

18 cm = 2 * radius * sin(120/2)

Simplifying:

9 cm = radius * sin(60)

Since sin(60) = √3/2, we can rewrite the equation as:

9 cm = radius * (√3/2)

To find the radius, we can rearrange the equation:

radius = (9 cm) / (√3/2)

Simplifying:

radius = (9 cm) * (2/√3)

radius = 6√3 cm

Finally, to find the diameter, we can multiply the radius by 2:

diameter = 2 * radius

diameter = 2 * 6√3 cm

diameter = 12√3 cm

Therefore, the diameter of the given circle is 12√3 cm.

Answer

The diameter of the given circle is 12√3 cm.