Question

Через точку М проведены касательные МК и МЕ к окружности с центром в точке О, где К и Е - точки касания, угол ОМК=30 градусов, угол МК= 6 см. Найдите длину хорды КЕ.

Answer 1

6 см.

 

МК = МЕ (по свойству касательных к окружности, проведенных из одной точки)

Углы ОМК и ОМЕ равны, т.к. отрезок ОМ соединяет центр окружности с общей точкой касательных. А значит, угол ОМЕ также равен 30 градусам, и треугольник МЕК равносторонний со стороной 6 см.

Answer 2

Problem Analysis

We are given a circle with center O, and two tangents MK and ME drawn to the circle. The angle OMK is 30 degrees and the length of MK is 6 cm. We need to find the length of chord KE.

Solution

To find the length of chord KE, we can use the properties of tangents and chords in a circle.

Let's denote the length of KE as x.

From the given information, we know that MK is a tangent to the circle. Therefore, the line segment MK is perpendicular to the radius drawn from the center of the circle to the point of tangency K. This means that the triangle OMK is a right triangle.

Using trigonometry, we can find the length of OK (which is the radius of the circle) using the given angle OMK and the length of MK.

Let's denote the length of OK as r.

In the right triangle OMK, we have:

sin(OMK) = MK / OK

Since we know the value of MK (6 cm) and the angle OMK (30 degrees), we can solve for OK:

sin(30) = 6 / OK

OK = 6 / sin(30)

Now that we have the length of OK, we can find the length of OE (which is also the radius of the circle) using the same trigonometric relationship:

sin(OME) = ME / OE

Since ME is also equal to MK (because both are tangents to the circle), we have:

sin(OME) = 6 / OE

Using the fact that the sine of an angle is equal to the sine of its complementary angle, we can rewrite the equation as:

sin(90 - OME) = 6 / OE

cos(OME) = 6 / OE

Now, we can use the cosine of the angle OME to find the length of OE:

OE = 6 / cos(OME)

Since OE is equal to OK (both are radii of the same circle), we have:

6 / sin(30) = 6 / cos(OME)

sin(30) = cos(OME)

Using the trigonometric identity sin(30) = cos(60), we can conclude that the angle OME is 60 degrees.

Now, we have a triangle OKE with angles OKE, OEK, and OKE (which is 60 degrees). We can use the Law of Cosines to find the length of KE:

KE^2 = OK^2 + OE^2 - 2 * OK * OE * cos(OKE)

Since OK = OE, we can simplify the equation to:

KE^2 = 2 * OK^2 - 2 * OK^2 * cos(OKE)

KE^2 = 2 * OK^2 * (1 - cos(OKE))

KE^2 = 2 * (6 / sin(30))^2 * (1 - cos(60))

KE^2 = 2 * (6 / sin(30))^2 * (1 - 0.5)

KE^2 = 2 * (6 / sin(30))^2 * 0.5

KE^2 = (6 / sin(30))^2

KE = 6 / sin(30)

Now, we can calculate the value of KE:

KE = 6 / sin(30) = 6 / 0.5 = 12 cm

Therefore, the length of chord KE is 12 cm.

Answer

The length of chord KE is 12 cm.