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

4.99 $

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

3 $

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

3.75 $

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

3 $