Question

Вася вырезал из картона треугольник и пронумеровал его вершины цифрами 1, 2, 3. Оказалось, что если Васин треугольник повернуть 12 раз по часовой стрелке вокруг его вершины под номером 1 на угол, равный углу при этой вершине, то он вернется в исходное положение. Если Васин треугольник повернуть 6 раз по часовой стрелке вокруг его вершины под номером 2 на угол, равный углу при этой вершине, то он вернется в исходное положение. Вася утверждает, что если повернуть его треугольник праз вокруг вершины под номером 3 на угол, равный углу при этой вершине, то он вернется в исходное положение. Какое минимальное п мог назвать Вася так, чтобы его утверждение было правдивым хотя бы для какого-то картонного треугольника? Число

Answer 1

Ответ:

Как по мне вроде бы 3. Если не правильно, сорян

Answer 2

To solve this problem, let's analyze the given information step by step.

Vasya cut out a triangle from cardboard and numbered its vertices with the digits 1, 2, and 3. He then made two claims about the triangle:

1. If he rotates the triangle 12 times clockwise around vertex 1, by an angle equal to the angle at that vertex, it will return to its original position. 2. If he rotates the triangle 6 times clockwise around vertex 2, by an angle equal to the angle at that vertex, it will return to its original position.

Now, Vasya claims that if he rotates the triangle around vertex 3, by an angle equal to the angle at that vertex, it will also return to its original position.

To find the minimum value for the angle at vertex 3 that makes Vasya's claim true, we need to consider the properties of triangles and rotational symmetry.

Properties of Triangles:

1. The sum of the interior angles of a triangle is always 180 degrees. 2. The angles opposite to equal sides of a triangle are equal.

Rotational Symmetry:

A shape has rotational symmetry if it can be rotated by a certain angle around a fixed point and still look the same.

Let's analyze the given information to find the minimum value for the angle at vertex 3.

Analysis:

1. Vasya claims that rotating the triangle 12 times around vertex 1 by an angle equal to the angle at that vertex will bring it back to its original position. - This means that the angle at vertex 1 must be a divisor of 360 degrees (12 * 30 degrees) for the triangle to return to its original position after 12 rotations. - The possible angles at vertex 1 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, and 330 degrees.

2. Vasya also claims that rotating the triangle 6 times around vertex 2 by an angle equal to the angle at that vertex will bring it back to its original position. - This means that the angle at vertex 2 must be a divisor of 360 degrees (6 * 60 degrees) for the triangle to return to its original position after 6 rotations. - The possible angles at vertex 2 are 60, 120, 180, 240, and 300 degrees.

Now, let's consider the possible angles at vertex 3 and check if Vasya's claim holds true for any of them.

Possible Angles at Vertex 3:

1. If the angle at vertex 3 is 30 degrees, the sum of the angles in the triangle would be 30 + 60 + 90 = 180 degrees. This satisfies the properties of a triangle, but it does not match the rotational symmetry condition for vertex 1 (12 rotations) or vertex 2 (6 rotations).

2. If the angle at vertex 3 is 60 degrees, the sum of the angles in the triangle would be 60 + 60 + 60 = 180 degrees. This satisfies the properties of a triangle, and it matches the rotational symmetry condition for vertex 2 (6 rotations). However, it does not match the rotational symmetry condition for vertex 1 (12 rotations).

3. If the angle at vertex 3 is 90 degrees, the sum of the angles in the triangle would be 90 + 60 + 30 = 180 degrees. This satisfies the properties of a triangle, but it does not match the rotational symmetry condition for vertex 1 (12 rotations) or vertex 2 (6 rotations).

4. If