Question

Из точки вне окружности проведены к ней две касательные. Радиус окружности равен 11см, сумма отрезков касательных равна 120 см. Найдите расстояние от данной точки до йентра окружности

Answer 1

Касательные проведенные из одной точки равны между собой =120/2=60, проводим радиусы перпендикулярные точкам касания и соединям точку вне окружности с центром , получаем два прямоугольных треугольника, в котором искомое расстояние =

= корень(касательная в квадрате+ радиус в квадрате) = корень(3600+121) =61

Answer 2

Problem Analysis

To solve this problem, we are given the following information: - Two tangents are drawn from a point outside the circle to the circle. - The radius of the circle is 11 cm. - The sum of the lengths of the two tangent segments is 120 cm.

We need to find the distance from the given point to the center of the circle.

Solution

Let's denote the lengths of the two tangent segments as x and y. According to the given information, the sum of the lengths of the two tangent segments is 120 cm. Therefore, we have the equation:

x + y = 120

We also know that the lengths of the tangent segments from an external point to a circle are equal. Therefore, we can denote both tangent segments as t.

Now, we can use the properties of tangents from an external point to a circle to find the lengths of the tangent segments.

The distance from the external point to the center of the circle can be found using the Pythagorean theorem. Let's denote this distance as d.

We can use the following equation to find d: d^2 = x * y

Calculation

Let's calculate the lengths of the tangent segments and the distance from the given point to the center of the circle.

Using the equation x + y = 120, we can solve for x and y.

x = 120 - y

Substitute x in the equation d^2 = x * y: d^2 = (120 - y) * y

Now, we need to find the value of y that maximizes d^2.

Maximizing d^2

To maximize d^2, we can use the concept of the arithmetic mean - geometric mean inequality. According to this inequality, the maximum value of d^2 occurs when x = y.

Substitute x = y in the equation x + y = 120: 2y = 120 y = 60

Substitute y = 60 in the equation d^2 = (120 - y) * y: d^2 = (120 - 60) * 60 d^2 = 3600 d = 60

Conclusion

The distance from the given point to the center of the circle is 60 cm.