Problem number 22. Thomas wants to shade two additional squares on the diagram shown so that the resulting pattern has a single axis of symmetry. And how many different ways can he do this? So another thing about geometry that we should remember is that a single square has four possible axes of symmetry shown in the diagram. So let's try all of these axes of symmetry and imagine reflecting the image about each of these axes of symmetry like this. Notice that in the first three cases the square already has four shaded squares and so that means that in these first three cases there is only one way to fill out the square so that that line is the line of symmetry. For example for the horizontal line there is only one way to complete the square so that horizontal line is a line of symmetry. And similarly we can do that for the top left to bottom right diagonal and the vertical or line of symmetry. However for the other line of symmetry from the bottom left to the top right that line only has three shaded squares which means that we have one extra shaded square that we need to place somewhere. And we notice that if we need to shade one square that must keep this line of symmetry a line of symmetry we have to shade it on the line itself. Because if we don't shade it on the line then we will need to shade its reflection across the line and that means we would need to shade two squares but we're only allowed to shade one. And so that means that we are allowed to shade any of the three squares on the line of symmetry itself as shown. We can shade that square that square or that square and all of them will be valid choices and so that means there are three ways to complete the diagram in that case. That means that the total number of ways is 1 plus 1 plus 1 plus 3 equals 6. So the answer is E, 6.

axis of symmetry

diagonal symmetry

symmetrical shading

square diagram

six ways