Question

Дуга радиус окружности которой равен 6 см и центральный угол которой 120 градусов свернута в окружность. Найдите радиус окружности. Чему равна хорда, стягиваемая этой дугой?

Answer 1

Найдем для начала длину дуги.Она равна 120*пи*6/180=4*пи

Затем найдем радиус из длины окружности. Т.к дуга свернута в окружность,то и длина окружности=длине дуги. 2*пи*r=4*пи.

r=2

После этого находим хорду через теорему косинусов. Ее можно найти как AB2=r2+r2+2*r*r*cos120=4+4+2*4*4*(cos60)=4+4+2*4*4*0.5=16

AB=4

Answer 2

Calculation of the Radius of the Circle

To find the radius of the circle, we can use the formula for the length of an arc of a circle. The formula is:

Length of Arc = (Central Angle / 360) * (2 * π * Radius)

In this case, the central angle is given as 120 degrees and the length of the arc is given as 6 cm. We can substitute these values into the formula and solve for the radius.

6 cm = (120 / 360) * (2 * π * Radius)

Simplifying the equation, we have:

6 cm = (1/3) * (2 * π * Radius)

Now, we can solve for the radius:

Radius = (3 * 6 cm) / (2 * π)

Using the value of π as approximately 3.14159, we can calculate the radius:

Radius ≈ (3 * 6 cm) / (2 * 3.14159) ≈ 2.86 cm

Therefore, the radius of the circle is approximately 2.86 cm.

Calculation of the Chord Length

To find the length of the chord, we can use the formula for the length of a chord in a circle. The formula is:

Chord Length = 2 * Radius * sin(Central Angle / 2)

In this case, the central angle is given as 120 degrees and the radius is approximately 2.86 cm. We can substitute these values into the formula and solve for the chord length.

Chord Length = 2 * 2.86 cm * sin(120 degrees / 2)

Using the sine function, we can calculate the chord length:

Chord Length ≈ 2 * 2.86 cm * sin(60 degrees)

Using the value of sin(60 degrees) as approximately 0.86603, we can calculate the chord length:

Chord Length ≈ 2 * 2.86 cm * 0.86603 ≈ 4.72 cm

Therefore, the length of the chord is approximately 4.72 cm.

Please note that the calculations provided are approximate due to the use of rounded values for π and sin(60 degrees).