Question bank: Lines, Line Segments, and Rays

During the construction of an art project promoted by a school, a group of 6th-grade students decided to create a large geometric mosaic in the shape of a chessboard in a nearby square. The mosaic will be composed of square ceramic pieces, all of the same dimension and side 'L'. To optimize the use of the pieces and ensure that all lines of the chessboard are continuous straight lines, the students decided to organize the pieces in such a way that each side of each square touches perfectly the side of two other squares. Considering an initial arrangement where a central piece is surrounded by three other pieces, forming a central 'X', and that the final project must have an even number of rows and an even number of columns of pieces, the students need to calculate the exact quantity of 'L' pieces needed to compose the mosaic. Knowing that the square has a rectangular shape and that the mosaic must occupy exactly half of the square's area, what is the minimum quantity of 'L' pieces that the students need to complete the mosaic considering that no piece is broken or cut?

In the city of Geometropolis, the streets were planned in such a way that they always meet at perfect right angles, forming a true tangle of straight lines. A citizen, Mr. Mathematician, needs to move from point A, located at the intersection of Matriz Street with Even Numbers Street, to point B, located at the intersection of Fundamental Theorem Street with Parallel Street. He wants to find the shortest path, which would necessarily be a straight line. However, the streets, although all straight, have different paving materials for construction reasons, and the cost of traveling through each of them is different. Matriz Street is considered a special road and has zero cost, Even Numbers Street has a unit cost of 5 monetary units per meter traveled, Fundamental Theorem Street costs 10 monetary units per meter, and Parallel Street, due to its luxurious finish, costs 15 monetary units per meter. Considering that Mr. Mathematician must follow a route consisting of exactly two straight segments, in addition to the initial stretch where he is, and that he can freely choose which sequence of streets to take, how should he plan his route to minimize the total cost of the journey, knowing that the length of the streets is 100 meters between each intersection and that he needs to travel a total of 300 meters?

Joana decided to create a drawing for her art project on a rectangular sheet of paper, using rulers and set squares. She intends to use parallel and perpendicular lines in her drawing. (a) Describe how Joana can use the ruler and set square to draw two parallel lines. (b) Now, explain how she can draw two perpendicular lines using the same tools. (c) Finally, guide Joana in constructing a rectangle using the skills acquired in the previous items.

If two or more line segments have the same length, they are called congruent.

Imagine you are an architect designing the layout of a classroom. This room will have a wall with two windows and a rectangular blackboard between them. The blackboard will be parallel to the windows and equidistant from each window. The room will also have two doors on the side walls. a) Using a ruler and a square, draw a sketch of the floor plan of this room, representing the windows, the blackboard, and the doors. b) Indicate on your sketch the parallel and perpendicular lines present in the drawing. c) Calculate the distance between the blackboard and each window, considering that each side of the wall is 10 meters long and the blackboard is 4 meters long.