Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem7

Video Transcription

Question number 7. Six points are marked on a grid as shown to the right. What geometric figure cannot have all of its vertices on some of these six points? So what I will do  is just go through the possibilities and show you if such a figure as described in the question  can be drawn or not. So first we have a square, and for the square we can connect the following  four vertices here. Each segment that I'm drawing in is the hypotenuse of a right triangle here, and so this will be a square. Now A is possible, so let's move on to B. Can we  have a trapezoid? And in the same picture, in red also, I'll draw in a trapezoid. So now we connect the remaining points like that, and so here we have a trapezoid. So  square here is A, trapezoid is B. Now can we have a parallelogram that is not a rhombus?  So let's make another copy here of the grid, and then we can draw now in blue a parallelogram  by connecting these two pairs, and then these two, and we have a parallelogram. The edges are not of equal length, clearly, so this is not a rhombus, and so we have here possibility  C like that. So let's move on. Can we have an obtuse triangle? Yes, we can. We can, for example, now in, let's do orange, we can connect these three points, and so the angle  here like so is obtuse, and then we can make this into a triangle like that, and we have  an angle in the triangle greater than 90 degrees, so that's an obtuse triangle, and so D here is also possible. So then by the process of elimination, an equilateral triangle cannot  be drawn, and if we try, in the original picture, we see that drawing sides like this possibly  makes us believe there is a equilateral triangle, but then the third side will not have the same length as the previous two, no matter where we draw this triangle. So this is not  a possibility. We can then try again with these vertices over here, and we see that the third side also will not have the same length as the previous two, and that pretty  much exhausts the possibilities that we have. So E is not possible. We cannot create here an equilateral triangle whose vertices all lie on some of these six points.

Video Summary

In the problem, six points are marked on a grid, and the task is to determine which geometric figure cannot have all its vertices on these points. The narrator explores several shapes: a square, trapezoid, non-rhombus parallelogram, and obtuse triangle. Each of these can be constructed with the given points. However, an equilateral triangle cannot be formed as the third side will not match the length of the other two, regardless of how the points are connected. Thus, an equilateral triangle is the figure that cannot have all its vertices on the given six points.

Keywords

geometric figures

equilateral triangle

grid points

vertices

shapes