Grades 9-10 Video Solutions 2013-Level 9&10 Video Solutions 2013 problem2

Video Transcription

Question number two, Mary drew six identical squares, each containing a shaded region, and here they are.  How many of the regions have a perimeter equal in length to the perimeter of one of the squares?  So to illustrate the problem, I have copied the first two squares and we will trace around the perimeter,  comparing the perimeter of the square to that of the shaded region to see what's going on.  So first of all, where they share a boundary,  I'll shade that in yellow,  and we have exactly the same length of both perimeters.  But when they do not share a boundary,  let's do that in red, for example here, I have to match that up with a boundary of the shaded region like so,  and see that these two lengths are the same, so so far I have created nothing extra.  And finally to complete the perimeter of the square, I draw in one more red line and that matches up with this boundary and I have the same perimeter  of the shaded region as of the outside square.  But moving on here, if I repeat the exercise, here is a piece  that matches up with the boundary of the shaded region and here is a piece of the boundary of the shaded region that matches with the outside boundary, but now moving  on to the overlapping pieces, here the outside  and the shaded region, they will share a perimeter of the same length, but if I shade this piece in, I now create a more perimeter. And likewise over here,  now the shaded region has a larger perimeter by adding in these two pieces and the overlap on the outside does not cause any problems, but these two inner pieces do.  And so to detect such a situation, what we can do is draw a line passing through the entire square and see how many intersections we have with the boundaries  of the shaded regions. And if we have more than two, then we run into trouble.  The outside two are fine and then anything else is extra. And so going through our examples,  we see that obviously the second one has this behavior and also the third one,  but all the other ones will only have two intersections, which is permitted. And so of the six, two do not work and we conclude that the answer to number two is C, four.

Video Summary

Mary drew six identical squares with shaded regions inside. The task was to determine how many shaded regions have a perimeter equal to the square's perimeter. By tracing the perimeters, it's observed that if a line drawn through the square intersects the shaded region's boundaries more than twice, the perimeters don't match. Upon examination, two of the six squares have this issue, leaving four squares where the shaded region's perimeter matches the square. Therefore, the answer to the problem is four.

Keywords

squares

shaded regions

perimeter

geometry

problem solving