Hi! I'm SO SORRY that this is REALLY late. I hope that I explain this question well. The Question I got is the chessboard question. THe first question talks about the diagonal lines of the small squares/rectangles.
A)Find the diagonal of all squares and rectanges possible in the first three rows. Then Arrange the squares/rectangles, from least to greatest by length of their diagonals.
As you can see, not all the little squares were "counted" because for example, row 3 and column 1 was already done by the first row, third column.(Do you get it?)
Now we need to find ALL the possible diagonals in the first three rows.
If it helps you draw a picture of the chess board and then do the math.
The picture(sorry that it's a bit blurry)on the bottom shows all the possible diagonal lines. IN ORDER from LEAST TO GREATEST
In order to get the diagonal of the squares/rectangle, you need to know that A squared + B squared = C squared (formula for a triangle)
The second part of this question is...
B) Consider the entire chessboard. When would you expect the square or rectangle to give a whole number diagonal?
The picture below shows how many squares/rectangles that give a whole number. When we did the three rows, we only found one.
Which was 3 squared + 4 squared = C squared
Then the square root of 10 = C squared
Then the answer was 5 = C squared
Then there's another diagonal with a whole number which was:
6 squared +8 squared = C squared
square root of 100 = C squared
Then you get the answer of 10 = C squared
If you look at the two diagonals that have a whole number as a diagonal, you could see that
6 squared +8 squared is four times as much as 3 squared + 4 squared. I don't really know how else I could explain part B of this question....so that's all I'm going to say...