GREATER CHALLENGES -II 3rd year of secondary education

 1. SQUARE GRIDS AND ANGLES IN A TRIANGLE In this window, as in the first window on the previous page, there is a square grid and a triangle drawn on it. However, this time instead of focusing on the shape's area we are going to study its angles. The electronic board works in a similar way to before. The question that you should aim to answer by the end of the unit is found in the last activity. 1.- Draw different triangles on the electronic board making sure that angle BAC is one of its acute angles in each case. For each triangle write down the size of angle A the lengths of the height and line AD and the ratio between these two lengths in your exercise book. Draw a table in your book to help you.

 2. THE CHALLENGE In this window you can move the vertices of the triangle to any of the points on the grid, using the mouse. You will need to carry out a fairly complex mathematical investigation at home in order to be able to answer the last two questions. So be patient and good luck! Big clue: If angle BAC is 60º, the relationship between the perpendicular height from point C (or from point B) and the distance from point A to the base of the perpendicular height is equal to the square root of 3. 2.- Repeat the above activity for different triangles that you draw on the electronic board changing vertices B and/or C, making sure that angle A is always acute (this angle is used as our point of reference). 3.- Use the information from the activities above to help you answer the following question: Is it possible to draw a triangle where angle BAC is 60º? 4.- Is it possible to draw equilateral triangles on a square grid where each of the triangle's vertices is located on a point on the grid? (Another way of asking this question is to ask if it is possible to make an equilateral triangle with a rubber band on a board with pins arranged like the points on a square grid, regardless of the number of pins per side). Josep Mª Navarro Canut ProyectoDescartes.org. Year 2013 