Question

коло вписане в рівнобедрений трикутник ділить його бічну сторону у відношенні 2:3 починаючи від вершини що протилежна основі знайдіть стороони трикутника якщо його периметр дорівнює 240

Answer 1

Ответ:

Согласно свойства касательных-если к окружности из одной точки проведены две касательные,то они равны

РВ=ВС=2

АР=АD=3

MC=DC=3

2X•2+6X•2=240

16X=240

X=240:16

X=15

AB=BC=(2+3)•15=75 ,как боковые стороны равнобедренного треугольника

Основание

АС=3•2•15=90

Р=75•2+90=240

Объяснение:

Answer 2

Problem Analysis

We are given that a circle is inscribed in an isosceles triangle, and the ratio of the divided side is 2:3. We need to find the lengths of the sides of the triangle if its perimeter is 240.

Solution

Let's assume that the base of the isosceles triangle is divided into two segments, with lengths 2x and 3x, where x is a common factor.

Since the triangle is isosceles, the other two sides are also equal in length. Let's call this length y.

To find the perimeter of the triangle, we can add up the lengths of all three sides:

Perimeter = 2x + 3x + 2y

Given that the perimeter is 240, we can set up the equation:

2x + 3x + 2y = 240

Simplifying the equation:

5x + 2y = 240

To solve for x and y, we need another equation. We can use the fact that the circle is inscribed in the triangle. The radius of the inscribed circle is equal to the perpendicular distance from the center of the circle to any side of the triangle.

Let's denote the radius of the inscribed circle as r. The formula for the radius of the inscribed circle in terms of the sides of the triangle is:

r = (s - a) / 2

Where s is the semi-perimeter of the triangle and a is the length of one side of the triangle.

Since the triangle is isosceles, we can use the lengths 2x and y for a. The semi-perimeter s is equal to (2x + 2y) / 2 = x + y.

Substituting the values into the formula, we get:

r = (x + y - 2x) / 2 = (y - x) / 2

We can also express the radius in terms of x and y using the Pythagorean theorem. The radius is the hypotenuse of a right triangle with legs x and y - x.

Using the Pythagorean theorem, we have:

r^2 = x^2 + (y - x)^2

Simplifying the equation:

r^2 = x^2 + y^2 - 2xy + x^2

r^2 = 2x^2 + y^2 - 2xy

Now we have two equations:

Equation 1: 5x + 2y = 240 Equation 2: r^2 = 2x^2 + y^2 - 2xy

We can solve these equations simultaneously to find the values of x and y.

Let's solve Equation 1 for y:

2y = 240 - 5x y = (240 - 5x) / 2

Substituting this value of y into Equation 2:

r^2 = 2x^2 + ((240 - 5x) / 2)^2 - 2x((240 - 5x) / 2)

Simplifying the equation:

r^2 = 2x^2 + (240 - 5x)^2 / 4 - x(240 - 5x) / 2

Now we have an equation in terms of x and r^2. We can solve this equation to find the values of x and r.

Once we have the values of x and r, we can calculate the values of y and the lengths of the sides of the triangle.

Note: Since we don't have the specific values of x and r, we cannot provide the exact lengths of the sides of the triangle. However, the above approach outlines the general method to solve the problem.

I hope this helps! Let me know if you have any further questions.