In a regular hexagon, a triangle is built, the vertices of which are the vertices of the hexagon, taken through one.

| February 9, 2021 | education

In a regular hexagon, a triangle is built, the vertices of which are the vertices of the hexagon, taken through one. Find the ratio of the perimeters of the hexagon and the resulting triangle.

In a regular hexagon, its sides divide the circle into six equal arcs, the degree measure of each of which is equal to: arc AB = 360/6 = 60. Then the degree measure of the arc MВС = 2 * 60 = 120. Then the value of the inscribed angle ВDM = 120 / 2 = 60, and the angle MAB = 180 – 60 = 120.

Let the length of the side of the hexagon be equal to X cm.The triangle ABM is equilateral from angles 1200, then by the cosine theorem, BM^2 = X^2 + X^2 – 2 * X * X * Cos120 = 2 * X^2 + X^2 = 3 * X^2.

BM = X * √3 cm.

The perimeter of the hexagon is: P6 = 6 * X cm.

The perimeter of the triangle is: P3 = 3 * X * √3 cm.

P6 / P3 = 6 * X / 3 * X * √3 = 2 / √3.

Answer: The ratio of the perimeters is 2 / √3.



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