Transformations on the plane

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Problem: Prove that all translations to vectors collinear to the vector \( \vec{a},(\vec{a} \neq 0) \), and all symmetries about axes, perpendicular to \( \vec{a} \), form a group.

5.2.1.1 Transformations on the plane

4.83 $

Problem: Prove that if a figure has two and only two axes of symmetry, these axes are perpendicular.

5.2.1.2 Transformations on the plane

2.9 $

Problem: Inside the triangle \( A B C \) an arbitrary point \( M \) is chosen. Find the perimeter of the triangle with vertices at the centroids of the triangles \( A B M, B C M \) and \( A C M \), if the perimeter of the triangle \( A B C \) is equal to \( 2 p \).

5.2.1.3 Transformations on the plane

3.62 $

Problem: Prove that if the line segment connecting the midpoints of the bases of the trapezoid forms equal angles with its sides, the trapezoid is isosceles.

5.2.1.4 Transformations on the plane

2.9 $