Question

Даны две окружности.Хорода АВ касается меньшей окружности и равна 8см.Найтите площадь кольца

Answer 1

Пусть О - центр окружностей.
Опустим из точки О перпендикуляр к АВ в точку касания С.
В треугольнике ОСВ катет ОС  равен радиусу r меньшей окружности, а гипотенуза ОВ равна радиусу R большей окружности.
По Пифагору СВ² = ОВ²-ОС². СВ² = R²-r².
СВ = (1/2)АВ = 3 см.
То есть 3² = R²-r².
Если обе части равенства умножить на π, то получим уравнение площади кольца: S = π(R²-r²) = 3²*π = 9π.

Answer 2

Problem Analysis

We are given two circles, where the chord AB is tangent to the smaller circle and has a length of 8 cm. We need to find the area of the ring (the region between the two circles).

Solution

To find the area of the ring, we need to subtract the area of the smaller circle from the area of the larger circle.

Let's denote the radius of the smaller circle as r and the radius of the larger circle as R.

We know that the chord AB is tangent to the smaller circle. This means that the line AB is perpendicular to the radius of the smaller circle at the point of tangency. Therefore, the length of AB is equal to twice the radius of the smaller circle (2r).

From the given information, we have AB = 8 cm, so 2r = 8 cm. Solving for r, we find that r = 4 cm.

To find the radius of the larger circle (R), we can use the Pythagorean theorem. The radius of the larger circle is the hypotenuse of a right triangle, where one side is r and the other side is the distance from the center of the larger circle to the midpoint of AB.

Since AB is a chord of the smaller circle, the distance from the center of the larger circle to the midpoint of AB is equal to the radius of the smaller circle (r). Therefore, the radius of the larger circle is R = r + r = 2r = 8 cm.

Now that we have the radii of both circles, we can calculate the areas of the circles and find the area of the ring.

The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.

The area of the smaller circle is A1 = πr^2 = π(4 cm)^2 = 16π cm^2.

The area of the larger circle is A2 = πR^2 = π(8 cm)^2 = 64π cm^2.

Finally, the area of the ring is the difference between the areas of the two circles:

Area of the ring = A2 - A1 = 64π cm^2 - 16π cm^2 = 48π cm^2.

Therefore, the area of the ring is 48π square centimeters.

Answer

The area of the ring formed by the two circles is 48π square centimeters.