For our Chopped Square puzzle, we attempted to find the side length of a square that was chopped up into 6 congruent rectangles.
Given a perimeter of 112 units for each of the six rectangles, what’s the length of each side of the square? Algebra steps in to help us out here.
Let’s first gather what we know.
Here is that square again:
First, the length of the small sides of each rectangle are equal to 1/6 of the length of the each side of the square. We know this because the rectangles are congruent, which means they have equal sides. If the shorter sides of the rectangles are all equal, and six of them make up the side of the square, then each small rectangle side is 1/6 the length of the square’s side. Let’s assign this 1/6 length a variable: x.
Second, the larger side of each rectangle is equal to the side of the square. This is immediately apparent from looking at the picture. Since we established above that the length of the square’s side is 6 times the length of the shorter rectangle side, we know that the larger rectangle side is also 6 times the length of the shorter one. Thus, the larger side can be expressed in terms of the shorter side: 6x.
The perimeter of a rectangle is length + length + width + width, and now we can fill those in with the terms we arrived at:
6x + 6x + x + x
We set that equal to 112, since that was the given perimeter of the rectangle:
6x + 6x + x + x = 112
Now, add together the like terms, which, in this case, is everything on the left side!
14x = 112
Divide 112 by 14, and we find out that x is equal to 8.
Remember that x is the length of the shorter rectangle side, and we set out to find the length of the square side, or the larger rectangle side. Since the larger rectangle side is 6x, all we have to do to get the answer is plug in 8 for x. That product, and therefore our answer, is 6 · 8 = 48!
We explore a similar problem in this tutorial here, to help you get the hang of it.
There are other ways to chop up the square and find out more dimensions from what little information we were initially given.
How did you solve the problem? Let us know in the comments!
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