Question

Угол С при вершине равнобедренного треугольника АВС равен 20, а основание — 10 см. Окружность радиуса АВ с центром в точке А пересекает боковые стороны в точках M и N. Найти MN.

Answer 1

Углы при основании (∡САВ=∡СВА) легко вычисляются: (180°-20°)/2 = 80°
треугольник NAB получится равнобедренным (с равными при основании BN углами): ∡ABN = ∡ANB = 80°,
угол NAB -центральный для окружности, ∡NAB = ∡АСВ = 20° 
∡СAN = ∡САВ - ∡NAB = 80° - 20° = 60° = ∡MAN -это тоже центральный угол для окружности, т.е. равнобедренный треугольник MAN является равносторонним, т.е. MN = r(=AB) = 10

Answer 2

Given Information

We are given that angle C at the vertex of an isosceles triangle ABC is 20 degrees, and the base AB is 10 cm. The circle with radius AB and center at point A intersects the side AC at point M and the side BC at point N. We need to find the length of MN.

Solution

To find the length of MN, we can use the properties of angles and segments in a circle.

First, let's consider triangle ABC. Since it is an isosceles triangle, we know that angles A and B are equal. Therefore, angle A is also 20 degrees.

Next, let's consider the circle with radius AB and center at point A. According to the properties of a circle, the angle formed by a chord and a tangent at the point of contact is equal to half the angle subtended by the chord at the center of the circle.

In our case, MN is a chord of the circle, and angle MAN is half of angle A. Therefore, angle MAN is 10 degrees.

Now, let's consider triangle AMN. We have angle MAN equal to 10 degrees, and angles AMN and ANM are equal because MN is a chord of the circle. Therefore, angles AMN and ANM are also 10 degrees each.

Since the sum of the angles in a triangle is 180 degrees, we can find angle AMN by subtracting the sum of angles MAN and ANM from 180 degrees:

angle AMN = 180 - (angle MAN + angle ANM) = 180 - (10 + 10) = 160 degrees

Now, we can use the Law of Sines to find the length of MN. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In triangle AMN, we can use the following equation:

MN / sin(AMN) = AN / sin(ANM)

Since angle AMN is 160 degrees and angle ANM is 10 degrees, we can rewrite the equation as:

MN / sin(160) = AN / sin(10)

We know that AN is equal to AB, which is 10 cm. Therefore, we can substitute the values into the equation:

MN / sin(160) = 10 / sin(10)

Now, we can solve for MN:

MN = (10 / sin(10)) * sin(160)

Using a calculator, we find that MN is approximately 3.82 cm.

Therefore, the length of MN is approximately 3.82 cm.