A large square is divided into 4 small congruent rectangles and a small square as shown. The areas of the large and small squares are 25 and 7, respectively. What is the length of a diagonal of a small rectangle?

To find the length of a diagonal of a small rectangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the small rectangle is congruent, so the lengths of the sides of the rectangle are the same.

Let's assume the length of each side of the small rectangle is x.

We can consider one of the small rectangles as a right-angled triangle, with the length of one side being x and the length of the other side being x/2 (half of the side length of the small square).

Using the Pythagorean theorem, we have:

x^2 = (x/2)^2 + (x/2)^2

Simplifying the equation:

x^2 = (x^2)/4 + (x^2)/4

x^2 = x^2/2

Multiplying both sides by 2 to eliminate the denominator:

2x^2 = x^2

We can cancel out x^2 from both sides of the equation:

2 = 1

This is not a valid equation, which means there is an error in our assumption.

It seems that there is no solution to this problem or there was an error in the given information.

To find the length of the diagonal of a small rectangle, we need to determine the dimensions of the rectangle first.

Let's assume the length of the small rectangle is L and the width is W.

From the given information, we know that the area of the large square is 25 and the area of the small square is 7.

Since the large square is divided into 4 small congruent rectangles, we can express the area of the large square as the sum of the areas of these rectangles:

25 = 4(LW) + (L^2)

We also know that the area of the small square is 7, so we can write another equation:

7 = L^2

By rearranging the second equation, we find that:

L = √7

Substituting this value of L into the first equation, we have:

25 = 4(√7 * W) + 7

Simplifying this equation:

25 - 7 = 4√7 * W

18 = 4√7 * W

Dividing both sides by 4√7:

18 / (4√7) = W

To find the length, we need L and W, so let's put the value of W back into the second equation:

L = √7

Now that we have the values of L and W, we can find the length of the diagonal of the small rectangle by applying the Pythagorean theorem:

Diagonal^2 = L^2 + W^2

Diagonal^2 = (√7)^2 + (18 / (4√7))^2

Diagonal^2 = 7 + (18^2 / (16 * 7))

Diagonal^2 = 7 + (324 / 112)

Diagonal^2 = 7 + 2.893

Diagonal^2 = 9.893

Taking the square root of both sides:

Diagonal = √9.893

Diagonal ≈ 3.146

Therefore, the length of the diagonal of a small rectangle is approximately 3.146.

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