Question

В окружгости с центром в точке O проведены две хорды AB и CD. Прямые AB и CD перпендикулярны и пересекаются в точке M, лежащей вне оружности. При этом AM=36, BM=6,CD=4корень из 46. Найти OM.

Answer 1

Обозначим r радиус окружности, точкой K
середину отрезка AB, а точкой L середину отрезка
CD. Поскольку треугольники AOB и COD
равнобедренные, OK и OL перпендикулярны AB
и CD соответственно.
Отрезок AB равен AM −BM =30. Четырёхугольник OKML
является прямоугольником, поэтому OL=AB/2+BM =21.
Из прямоугольного треугольника ODL находим
r =√OL2 +DL2 =25.
Из прямоугольного треугольника OKB находим
OK =√r2 −KB2 =20.
Из прямоугольного треугольника OKM находим

OM =√OK2 +KM2 =29.
Ответ: 29.

 

Answer 2

Problem Analysis

We are given that two perpendicular chords, AB and CD, intersect at point M, which lies outside the circle. We are also given that AM = 36, BM = 6, and CD = √46. We need to find OM.

Solution

To solve this problem, we can use the properties of perpendicular chords and the power of a point.

Let's denote the center of the circle as O.

From the given information, we know that AM = 36, BM = 6, and CD = √46.

Using the power of a point theorem, we can determine that AM * BM = CM * DM.

Since AM * BM = 36 * 6 = 216, we can substitute this value into the equation:

216 = CM * DM.

We also know that CD = √46, so CM * DM = √46.

Therefore, √46 = CM * DM.

Since CM = DM, we can rewrite the equation as CM^2 = √46.

Taking the square root of both sides, we get CM = √√46.

Now, let's consider the right triangle OCM.

We know that OM is the hypotenuse of this triangle, and CM is one of the legs.

Using the Pythagorean theorem, we can write:

OM^2 = CM^2 + CO^2.

Since CM = √√46, we can substitute this value into the equation:

OM^2 = (√√46)^2 + CO^2.

Simplifying, we get:

OM^2 = √46 + CO^2.

We also know that CO is the radius of the circle, which is equal to the distance from O to any point on the circle.

Since AB and CD are perpendicular chords, the distance from O to M is equal to the radius of the circle.

Therefore, OM = CO.

Substituting this into the equation, we get:

OM^2 = √46 + OM^2.

Simplifying, we find:

2OM^2 = √46.

Dividing both sides by 2, we get:

OM^2 = √46 / 2.

Taking the square root of both sides, we find:

OM = √(√46 / 2).

Therefore, OM is equal to √(√46 / 2).

Answer

The value of OM is equal to √(√46 / 2).