Square in a square problem
In the figure, the bigger square has side x meters and the smaller square has side y meters, where x and y are whole number. If the area of the shaded region is 49 square meters, what is its perimeter?
The answer is 100.
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1 solution
The smaller square are completely inside the bigger square and it doesn't touch the bigger square as shown in figure. Note that, there are gaps from the FOUR sides around the smaller square. In this case the perimeter is 4 * y + 4 * x .
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Sorry for the figure. Two sides (left and bottom) of the smaller square must touch the sides of the bigger square.
Why did you equate (x+y)(x-y) to 49*1? What if I change the area to 36 square meters?
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He equated (x+y)(x-y) to 49 x 1 like this: the area of the shaded region is A = x 2 − y 2 which factors as x 2 − y 2 = ( x + y ) ( x − y ) . Since you told us it is 4 9 , he equated ( x + y ) ( x − y ) = 4 9 . Then since we are dealing with positive whole numbers, and the only divisors of 4 9 are 4 9 , 7 , 1 , the only valid solution is (x+y) = 49 and (x-y) = 1 (if x+y = 7, then y=0, so it is not valid.)
The shaded region is x2 – y2 = 49 , (x+y)(x – y) = 49 x 1, x+y=49, x – y =1, So x=25, y=24. The perimeter of the shaded region is 4(x – y) + 4y = 4x = 100.