A square ABCD is inscribed in a circle. The circle is folded to form a semi-sircle, and
another square EFGH is inscribed in this semi-cirlcle. Find the ratio of the area of the
square EFGH to the area of the square ABCD.
To find the ratio of the areas of the squares, we need to find the dimensions of the squares EFGH and ABCD.
Let's assume that the side length of square ABCD is "a". Since square ABCD is inscribed in a circle, the diagonal of square ABCD is equal to the diameter of the circle. The diagonal of a square can be found using the Pythagorean theorem:
Diagonal^2 = Side^2 + Side^2
D^2 = a^2 + a^2
D^2 = 2a^2
D = √(2a^2) = √2 * a
Now, let's consider the semi-circle formed by folding the circle. The diameter of the semi-circle is equal to the diagonal of square ABCD, which we found to be √2 * a. The radius of the semi-circle is half of the diameter, so:
Radius = (√2 * a) / 2 = √2 * a / 2
The diagonal of the square EFGH is equal to the diameter of the semi-circle, so it is also √2 * a. Therefore, the side length of square EFGH is:
Side length of EFGH = (√2 * a) / √2 = a
Now let's find the area of the two squares:
Area of square ABCD = a^2
Area of square EFGH = (a)^2 = a^2
The ratio of the area of square EFGH to the area of square ABCD is:
(a^2) / (a^2) = 1
Therefore, the ratio of the area of square EFGH to the area of square ABCD is 1:1.
To find the ratio of the area of square EFGH to the area of square ABCD, we can start by considering the properties of the two shapes.
Let's begin by understanding the relationship between the inscribed square and the circle. Since the square is inscribed in the circle, it means that its four vertices (A, B, C, and D) lie on the circumference of the circle.
Next, we'll analyze how the circle is folded to form a semicircle. After the fold, the two remaining vertices of square ABCD (E and F) will come together with the center of the circle (O), forming the semi-circle. Additionally, the other two vertices of square EFGH (G and H) will lie on the arc of the semi-circle.
To calculate the ratio of the areas, we can consider that the area of a square is proportional to the square of its side length. Hence, we need to determine the relationship between the side lengths of the two squares.
Since the vertices of square ABCD lie on the circumference of the circle, it means that the diagonal of the square is equal to the diameter of the circle. Let's represent the side length of square ABCD as "a" and the diameter of the circle as "d."
Using the Pythagorean theorem, we can find the relationship between "a" and "d."
a² + a² = d²
2a² = d²
a² = (1/2)d²
a = √(1/2)d
Now, let's consider square EFGH, which is inscribed in the semicircle. Since its neighboring vertices (E and F) come together with the center of the circle (O), we can conclude that the side length of square EFGH is equal to the radius of the circle (which is also half the diameter).
Therefore, the side length of square EFGH can be represented as "r" (which is half the value of "d").
r = (1/2)d
Now, we can find the ratio of the areas of the two squares.
(area of square EFGH) / (area of square ABCD) = (r²) / (a²)
= ((1/2)d)² / (√(1/2)d)²
= (1/4)d² / (1/2)d
= (1/4)d / (1/2)
= (1/4)d * (2/1)
= 1/2
Therefore, the ratio of the area of square EFGH to the area of square ABCD is 1/2.