Question

Бічна сторона рівнобедреного трикутника ділиться точкою дотику вписаного кола у відношенні 4 : 3, рахуючи від вершини кута при основі. Знайдіть бічну сторону трикутника, якщо його периметр дорівнює 132 см

Answer 1

The Side Length of an Isosceles Triangle Divided by the Tangent Point of the Inscribed Circle

To find the side length of the triangle, we need to use the given information about the ratio of the division of the side by the tangent point of the inscribed circle. Let's break down the problem step by step.

1. Let's assume that the length of the base of the isosceles triangle is x. 2. According to the given information, the side length of the triangle is divided by the tangent point of the inscribed circle in a ratio of 4:3, counting from the base angle vertex. This means that the length of the shorter segment is 4a and the length of the longer segment is 3a. 3. The perimeter of the triangle is given as 132 cm. Since it is an isosceles triangle, the sum of the lengths of the two equal sides is equal to the base length. Therefore, the perimeter can be expressed as: x + 2a + 2a = 132. 4. Simplifying the equation, we get: x + 4a = 132.

Now, we have two equations: - x + 4a = 132 (Equation 1) - 4a + 3a = x (Equation 2)

To find the values of x and a, we can solve these equations simultaneously.

Let's solve the equations:

From Equation 2, we can simplify it to: 7a = x.

Substituting this value of x in Equation 1, we get: 7a + 4a = 132.

Simplifying further, we have: 11a = 132.

Solving for a, we get: a = 12.

Substituting this value of a in Equation 2, we can find x: x = 7a = 7 * 12 = 84.

Therefore, the base length of the isosceles triangle is 84 cm.

To summarize: - The base length of the isosceles triangle is 84 cm. - The shorter segment of the side length is 4a = 4 * 12 = 48 cm. - The longer segment of the side length is 3a = 3 * 12 = 36 cm.

Please let me know if you need any further assistance!