Question bank: Translations of Plane Figures

During an Arts class, students were challenged to create colorful geometric patterns using paper triangles. João decided to create a pattern using an initial triangle and applying a sequence of translations to create a more complex design. João's initial triangle has sides measuring 6 cm, 8 cm, and 10 cm, and he wants to translate this triangle horizontally to the right, always maintaining the same orientation, so that the translated triangles fit perfectly, sharing one side. If João intends to create a pattern that has a total of 5 triangles after the translations, what is the horizontal distance he should move each triangle to form the complete pattern? Consider that there is no overlap between the triangles and that João started drawing the pattern with the lower left vertex of the initial triangle coinciding with the lower left vertex of the paper.

During an interdisciplinary activity of Mathematics and Arts, the 7th grade students were challenged to create a mosaic using the hydraulic tile technique, common in art deco style constructions. For the mosaic, each student received pieces in the shape of an equilateral triangle with a side length of 6 centimeters. After completing the mosaic, one of the students noticed that by translating a set of three tiles, forming a parallelogram common in various art deco designs, the resulting figure was also a piece of the mosaic. Considering the original position of the first set of three tiles as the reference point, and knowing that the second set of three tiles was translated 2 centimeters to the right and 4 centimeters upwards, determine the new position of the set of tiles after the translation.

During an interdisciplinary project in mathematics and arts, the 7th grade students decided to create an installation in the schoolyard that explored translation concepts in the Cartesian plane, using geometric figures printed on acrylic plates. The initial proposal was to compose a design with equilateral triangles that, when translated in a specific direction, formed interesting symmetrical patterns. The group had already designed the first stage of the installation, which consisted of a base formed by seven of these juxtaposed triangles, forming a regular hexagon, as shown in Figure 1. Now, the students need to determine the coordinates of the vertices of a second hexagon, which will be obtained from the first one through a translation that moves each vertex 4 units to the right and 2 units up. Considering that the original hexagon is centered at the origin of the Cartesian plane and that the vertices are arranged clockwise from the point (0, 3√3), which is the upper right vertex of the original hexagon, what will be the new position of the vertex that was initially at (-3, 3√3)?

João was observing a mosaic on the floor and noticed that some of its repetitive patterns were formed using translations. Given that one of the patterns is a rectangle 4 cm wide and 6 cm high, what would be the final coordinate of point A(2, 3) after it is translated 3 times to the right and 2 times upwards?

A video game company is developing a new maze game, and the designer needs to create a maze where the main character is translated horizontally and vertically along the path. Considering that the character starts at position (3, 4) and is translated 6 units to the right and 8 units up, what will be the new position of the character on the Cartesian plane?