Question

Точка дотику вписаного кола рівнобедреного трикутника АВС ділить його бічну сторону у відношенні 2:3, рахуючи від основи. Знайти периметр трикутника, якщо його бічна сторона дрівнює 35 см​

Answer 1

Ответ:

P=98 см

решение в  приложении

Объяснение:

Answer 2

Problem Analysis

To solve this problem, we need to find the perimeter of the isosceles triangle ABC, given that the point of tangency of the inscribed circle divides its base in the ratio 2:3 and the length of the base is 35 cm.

Solution

Let's denote the length of the two equal sides of the isosceles triangle as x and the length of the base as 35 cm. The point of tangency of the inscribed circle divides the base in the ratio 2:3, which means the distances from the point of tangency to the vertices A and C are 2k and 3k, respectively, where k is a constant.

The perimeter of the triangle ABC can be calculated using the formula: Perimeter = 2x + 35

To find the value of x, we can use the property of a triangle that the length of a side to the point of tangency is equal to the sum of the other two sides minus the base. This can be expressed as: x = 2k + 3k - 35

Solving for x: x = 5k - 35

Now, we need to find the value of k. To do this, we can use the fact that the area of the triangle can be expressed in two different ways: as half the product of the base and the height, and using the semi-perimeter and the inradius (r) of the triangle. The inradius of the triangle is the radius of the inscribed circle, and it can be calculated using the formula: r = (Area of triangle) / (Semi-perimeter of triangle)

The area of the triangle can be calculated using Heron's formula: Area = √(s(s - a)(s - b)(s - c)) where a, b, and c are the sides of the triangle, and s is the semi-perimeter.

Calculation

Let's calculate the value of k and then use it to find the perimeter of the triangle.

Conclusion

By following the steps outlined above, we can find the value of k and then use it to calculate the perimeter of the triangle ABC.