Question

Хорда окружности, равная 8 см, отсекает от окружности дугу 90 градусов ,Найдите расстояние от центра окружности до этой хорды .

Answer 1

Сединим центр окружности с концами хорды. получим тр-к ABC
в нем из точки О опустим перпендикуляр на хорду AB, это будет высотой OK нашего тр-ка
угол AOB = 90 градусов (центр угол AOB опирается на дугу =90гр)
OA=OB=r - след-но треугольник AOB прямоугольный равнобедренный
углы при основании 45 гр.
отсюда OK высота и медиана тр-ка, значит OK=KB=1/2AB=1/2*8=4

Answer 2

Finding the Distance from the Center of a Circle to a Chord

To find the distance from the center of a circle to a chord, we can use the following formula:

Distance from the center to the chord = √(r^2 - (d/2)^2)

Where: - r is the radius of the circle - d is the length of the chord

In this case, the chord has a length of 8 cm and it cuts off a 90-degree arc from the circle. We need to find the distance from the center of the circle to this chord.

Let's calculate the distance using the formula mentioned above.

Calculation

Given: - Length of the chord (d) = 8 cm - Angle of the chord (θ) = 90 degrees

To find the radius (r), we can use the formula:

r = (d/2) / sin(θ/2)

Substituting the values:

r = (8/2) / sin(90/2) r = 4 / sin(45) r = 4 / 0.7071 r ≈ 5.6569 cm

Now, we can calculate the distance from the center to the chord using the formula:

Distance from the center to the chord = √(r^2 - (d/2)^2)

Substituting the values:

Distance = √((5.6569)^2 - (8/2)^2) Distance = √(31.9996 - 16) Distance = √15.9996 Distance ≈ 3.9999 cm

Therefore, the distance from the center of the circle to the chord is approximately 3.9999 cm.

Please note that the calculations are approximate due to rounding.

Conclusion

The distance from the center of the circle to the chord, which has a length of 8 cm and cuts off a 90-degree arc, is approximately 3.9999 cm.