Question

К окружности, вписанной в равносторонний треугольник со стороной 6, проведена касательная, пересекающая две его стороны. Найти периметр отсеченного треугольника/Спасибо.

Answer 1

Периметр в 3 раза меньше т.к. Высота в треугольнике равна 3 радиусам вписанной окружности

Answer 2

Perimeter of the Cut-off Triangle in an Equilateral Triangle with Inscribed Circle

To find the perimeter of the cut-off triangle, we need to determine the lengths of its sides. Let's break down the problem step by step.

1. Identify the given information: - The equilateral triangle has a side length of 6 units. - A tangent is drawn to the inscribed circle, intersecting two sides of the equilateral triangle.

2. Find the radius of the inscribed circle: - In an equilateral triangle, the radius of the inscribed circle is given by the formula: radius = (side length * √3) / 6. - Substituting the given side length of 6 units into the formula, we can calculate the radius of the inscribed circle.

   radius = (6 * √3) / 6
   

Simplifying the expression:

   radius = √3
   

Therefore, the radius of the inscribed circle is √3 units.

3. Determine the length of the segment cut off by the tangent: - When a tangent is drawn to a circle from an external point, the length of the segment cut off by the tangent is equal to the square root of the product of the lengths of the two external segments. - In this case, the tangent intersects two sides of the equilateral triangle, dividing them into two segments. - Since the triangle is equilateral, the two segments are equal in length. - Let's denote the length of each segment as x.

   x = √(6 * √3)
   

Simplifying the expression:

   x = √(6√3) = √6 * √√3 = √6 * (3^(1/4))
   

Therefore, the length of each segment cut off by the tangent is √6 * (3^(1/4)) units.

4. Calculate the perimeter of the cut-off triangle: - The cut-off triangle is formed by the tangent and the two segments cut off by it. - Since the two segments are equal in length, the perimeter of the cut-off triangle is the sum of the lengths of the two segments and the length of the tangent.

   Perimeter = 2 * x + 2 * radius
             = 2 * (√6 * (3^(1/4))) + 2 * √3
   

Simplifying the expression:

   Perimeter = 2√6 * (3^(1/4)) + 2√3
   

Therefore, the perimeter of the cut-off triangle is 2√6 * (3^(1/4)) + 2√3 units.

In conclusion, the perimeter of the cut-off triangle in the equilateral triangle with an inscribed circle, where a tangent intersects two sides, is 2√6 * (3^(1/4)) + 2√3 units.