Alicia has a flat platter shaped like a square with a semicircle of diameter 10 inches on each edge, as pictured. What is the number of inches in the side length of the smallest square ABCD that can contain the platter?
I drew a diagram so far, but I don't know what to do next.
To find the side length of the smallest square that can contain the platter, we need to determine the diagonal of the platter and use it as the side length of the square.
Let's first analyze the diagram you drew. You correctly identified that the platter has a square shape with a semicircle on each edge. We can label the four corners of the square as A, B, C, and D.
Now, if we draw the diagonals of the square inside the platter, they will divide the platter into four congruent triangles. Let's label the points where the diagonals intersect the semicircles as E, F, G, and H.
Since each semicircle has a diameter of 10 inches, the radius (denoted as r) of each semicircle is 5 inches.
From point E to either A or B, the distance is equal to r, which is 5 inches.
Now, consider the right triangle EAD. One leg is the radius r, and the hypotenuse is the diagonal of the platter. Using the Pythagorean theorem, we can find the length of the diagonal:
(diagonal)^2 = (r)^2 + (r)^2
(diagonal)^2 = 5^2 + 5^2
(diagonal)^2 = 50 + 25
(diagonal)^2 = 75
Taking the square root of both sides:
diagonal = √75
diagonal ≈ 8.66 inches (rounded to two decimal places)
The diagonal of the platter is approximately 8.66 inches. Since the smallest square that can contain the platter has a side length equal to its diagonal, the side length of the smallest square ABCD would also be approximately 8.66 inches.
Therefore, the number of inches in the side length of the smallest square ABCD that can contain the platter is approximately 8.66 inches.
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