Question

В РАВНОБЕДРЕННОМ ТРЕУГОЛЬНИКЕ УГОЛ ПРИ ВЕРШИНЕ ОСНОВАНИЯ РАВЕН 2Х. НАЙТИ ОТНОШЕНИЕ ДЛИН РАДИУСОВ ВПИСАННОЙ И ОПИСАННОЙ ОКРУЖНОСТЕЙ.

Answer 1

АВС равнобедренный Δ

А=С=2х

ВД=а -высота из В

АВ=ВС=а/sin2x

AC=2a/tg2x

r=S/p=0,5*ВД*АС/(АВ+ВС+СА)=(а²/tg2x)/(2a/sin2x+2a/tg2x)=а/(2+2cos2x)=а/cos²x

R=АВ*ВС*СА/4(0,5*ВД*АС)=АВ*ВС/(2ВД)=а/2sin²2x

r/R=2sin²2x/cos²x

 

 

 

 

Answer 2

Finding the Ratio of Radii in an Isosceles Triangle

To find the ratio of the radii of the inscribed and circumscribed circles in an isosceles triangle, we need to consider the properties of the triangle and apply some trigonometry.

Let's denote the base angle of the isosceles triangle as x. According to the given information, the angle at the vertex of the base is equal to 2x.

To proceed, we can use the Law of Sines and the Law of Cosines to relate the sides and angles of the triangle.

Law of Sines

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 this case, we can use it to relate the sides of the triangle.

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. We can use it to find the length of the base of the triangle.

Now, let's proceed with the calculations.

1. Using the Law of Sines, we can relate the sides of the triangle: - Let a be the length of the base of the triangle. - Let r1 be the radius of the inscribed circle. - Let r2 be the radius of the circumscribed circle.

According to the Law of Sines, we have the following relationship: a / sin(x) = 2r1 / sin(2x) = 2r2 / sin(180 - 3x).

2. To find the length of the base a, we can use the Law of Cosines: - The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite side c, the following relationship holds: c^2 = a^2 + b^2 - 2ab * cos(C).

In our case, we have an isosceles triangle, so a = a, b = a, and C = 2x. Substituting these values into the Law of Cosines, we get: a^2 = a^2 + a^2 - 2a * a * cos(2x).

Simplifying the equation, we find: a^2 = 2a^2 - 2a^2 * cos(2x).

Solving for a, we get: a = 2a * cos(2x).

Dividing both sides by 2a, we find: 1 = cos(2x).

Since cos(2x) = 1 when 2x = 0, we have: 2x = 0.

Therefore, x = 0.

However, this solution is not valid because the angle at the vertex of the base cannot be zero. Therefore, there is no solution for this problem.

In conclusion, there is no valid solution for finding the ratio of the radii of the inscribed and circumscribed circles in an isosceles triangle with an angle of 2x at the vertex of the base.