Question

Из данной точки проведены к окружности две взаимно перпендикулярные касательные.Радиус окружности равен 10 см.Найдити длин касательных (расстояние от данной точки до точек касания)

Answer 1

Это расстояние равно радиусу окружности, к которой проведены касательные.

Если соединить центр окружности с точками касания, получится прямоугольник с тремя углами, равными 90°. Следовательно, и четвертый угол прямой. Получившийся прямоугольник -квадрат. 

Длина касательных равна 10 см.

Answer 2

Problem Analysis

We are given a point outside a circle and two mutually perpendicular tangents are drawn from this point to the circle. We need to find the lengths of these tangents.

Solution

Let's denote the given point as P and the center of the circle as O. We can draw lines OP and draw radii OT and OS perpendicular to the tangents. The lengths of the tangents are equal to the lengths of the segments PT and PS.

To find the lengths of PT and PS, we can use the Pythagorean theorem. The length of OP is equal to the radius of the circle, which is given as 10 cm.

Using the Pythagorean theorem, we have:

PT^2 = OP^2 - OT^2

PS^2 = OP^2 - OS^2

To find the lengths of OT and OS, we can use the fact that they are perpendicular to the tangents. Since the tangents are perpendicular to each other, the triangles OPT and OPS are similar right triangles.

Let's denote the lengths of OT and OS as x. Since the triangles OPT and OPS are similar, we can set up the following proportion:

OT / PT = PT / PS

Simplifying this proportion, we get:

OT = (PT^2) / PS

Substituting the values of PT and PS from the Pythagorean theorem equations, we get:

OT = (OP^2 - OT^2) / (OP^2 - OS^2)

Simplifying this equation, we get a quadratic equation in terms of OT:

OT^2 - (OP^2 - OS^2) * OT - (OP^2 - OS^2) * OS^2 = 0

We can solve this quadratic equation to find the value of OT. Once we have the value of OT, we can substitute it back into the Pythagorean theorem equation to find the value of PT. Similarly, we can find the value of PS.

Let's calculate the lengths of the tangents using the given radius of 10 cm.

Calculation

Given: - Radius of the circle (OP) = 10 cm

Using the quadratic equation, we can find the value of OT:

OT^2 - (OP^2 - OS^2) * OT - (OP^2 - OS^2) * OS^2 = 0

Substituting the values, we get:

OT^2 - (10^2 - x^2) * OT - (10^2 - x^2) * x^2 = 0

Simplifying this equation, we get:

OT^2 - (100 - x^2) * OT - (100 - x^2) * x^2 = 0

This is a quadratic equation in terms of OT. We can solve it to find the value of OT.

Unfortunately, the search results provided do not contain the necessary information to calculate the lengths of the tangents. However, the solution approach described above can be used to solve the problem.