A square and four congruent rectangles are arranged in a larger square. The smaller square and each of the rectangles have the same perimeter.Determine the ratio of the length of a side of the larger square to the length of side of the smaller square.
Let A is the length of a side of the larger square, a - of the smaller square.
The length of the smaller side of
a rectangle is (A-a)/2, the length of
the bigger side is (A-a)/2+a.
4a=2((A-a)/2+(A-a)/2+a)
4a=2A
A/a=2/1
Let's assume the side length of the smaller square is "s", and the side length of each rectangle is "r".
The perimeter of the smaller square is equal to 4s, and the perimeter of each rectangle is equal to 2(r+s).
Since there are four rectangles and one smaller square, the total perimeter of the rectangles is equal to 4 times the perimeter of a rectangle, which is 4(2(r+s)) = 8(r+s).
Given that the perimeter of the smaller square is equal to the perimeter of the rectangles, we can set up an equation:
4s = 8(r+s)
Simplifying this equation, we can divide both sides by 4:
s = 2(r+s)
Now, let's substitute the side length of a rectangle into the equation:
s = 2(r+s)
s = 2(r+2r)
s = 2(3r)
s = 6r
Therefore, the ratio of the side length of the larger square to the side length of the smaller square is:
Larger square side length / Smaller square side length = 6r / r = 6.
Thus, the ratio is 6:1.
To solve this problem, let's assume the side length of the smaller square is "s".
Since the perimeter of the smaller square is equal to the combined perimeter of the four rectangles, we can set up the following equation:
4s = 2*(2s + x) + 2*(2s + y),
where "x" is the width of one rectangle and "y" is the length of one rectangle.
Since the four rectangles are congruent and we have four of them, we can assume the length and width are equal for each rectangle. So x = y = a.
Now we can simplify the equation:
4s = 2*(2s + a) + 2*(2s + a),
4s = 4s + 2a + 2a,
0 = 4a.
Since a cannot be zero (as it represents the side length of the rectangle), it means that a = 0.
If a = 0, it means that the four rectangles do not exist, and we are left only with the smaller square. Therefore, the ratio of the length of a side of the larger square to the length of a side of the smaller square is 1:1.
In other words, the side length of the larger square is the same as the side length of the smaller square.