Question

Центр окружности, вписанной в равнобедренный треугольник с основанием 48 см,делит высоту, проведенную к основанию, в отношении 3:5, считая от основания. Найдите радиус окружности, описанной около треугольника. Помогите пожалуйста,буду очень признательна,заранее огромное спасибо!

Answer 1

Пусть боковая сторона равна b.

Центр впиcанной окружности находится на пересечении биссектрис.

Биссектриса угла делит противолежащаю сторону соотвественно в отношении равным отношению сторон

b/ (a/2)=5/3

a=48

a/2=48/2=24

b=5*24/3=40

 

Радиус описанной окружности вокруг равнобедренного треугольника равен

R=a*b*b/(4*a/2*корень(b^2-(a/2)^2))=

=48*40*40/ (2*48*корень(40^2-24^2))=25

ответ: 25 см

Answer 2

Problem Analysis

We are given a triangle with a base of 48 cm, and the center of the circle inscribed in the triangle divides the height, drawn to the base, in the ratio of 3:5. We need to find the radius of the circumcircle of the triangle.

Solution

Let's denote the base of the triangle as a and the height as h. The center of the inscribed circle divides the height in the ratio of 3:5, so we can write:

h1 = (3/8)h (1)

We can also use the formula for the area of a triangle:

S = (1/2)ah (2)

Since the triangle is isosceles, we can also write:

S = (1/2)bh (3)

where b is the length of the other two sides of the triangle.

From equations (2) and (3), we can equate the two expressions for the area of the triangle:

(1/2)ah = (1/2)bh (4)

Simplifying equation (4), we get:

a = b (5)

Now, let's find the values of a and h in terms of the given information.

From equation (5), we know that the length of the other two sides of the triangle is also 48 cm.

Using equation (1), we can find h1:

h1 = (3/8)h = (3/8)(48) = 18 cm

Using equation (2), we can find S:

S = (1/2)ah = (1/2)(48)(18) = 432 cm^2

Now, let's find the radius of the circumcircle of the triangle.

The area of a triangle can be calculated using the formula:

S = (abc)/(4R) (6)

where a, b, and c are the lengths of the sides of the triangle, and R is the radius of the circumcircle.

Since the triangle is isosceles, we can write:

S = (a^2h)/(4R) (7)

Substituting the values we know, we get:

432 = (48^2h)/(4R) (8)

Simplifying equation (8), we get:

R = (48^2h)/(4 * 432) = (48h)/36 = (4/3)h (9)

Substituting the value of h we found earlier, we get:

R = (4/3)(18) = 24 cm

Therefore, the radius of the circumcircle of the triangle is 24 cm.

Answer

The radius of the circumcircle of the triangle is 24 cm.

Explanation

We can find the radius of the circumcircle of a triangle by using the formula R = (4/3)h, where h is the height of the triangle. In this case, the height of the triangle is 18 cm, so the radius of the circumcircle is 24 cm