Another four-leaf clover also consists of 4 coplanar circles. The large circular leaves are externally tangent to each other, as well as to each of the smaller circular leaves, which are also congruent to one another. The radius of the large and small circular leaves is 1 1/2 inches and 1 inch, respectively. What is the area of a rhombus formed such that each of its vertices is also the center of one of the four circular leaves?
A man distributed 50 circular plates radius 7cm. To avoid rusting of them he electroplated them on both the sides at the rate of paise 50 per sq.cm. Find the expenditure of electrolpating
To find the area of the rhombus, we need to determine the length of one of its sides.
First, let's consider a side of the rhombus formed by two large circular leaves. Since the radius of the large circular leaves is 1 1/2 inches, the distance between the centers of two adjacent large circular leaves (which is also the length of one side of the rhombus formed by them) is 2 times the radius, or 2 * 1 1/2 inches = 3 inches.
Now, let's consider a side of the rhombus formed by a large and a small circular leaf. The radius of the small circular leaves is 1 inch, so the distance between the center of a large circular leaf and a small circular leaf (which is also the length of one side of the rhombus formed by them) is the sum of the radii, or 1 1/2 inches + 1 inch = 2 1/2 inches.
Therefore, the length of one side of the rhombus is either 3 inches (formed by two large circular leaves) or 2 1/2 inches (formed by a large and a small circular leaf).
Since a rhombus has all four sides equal in length, we can conclude that the length of one side of the rhombus is 2 1/2 inches.
Next, let's find the area of the rhombus.
The formula to find the area of a rhombus is A = (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals.
In this case, since all four vertices of the rhombus are the centers of the circular leaves, the diagonals of the rhombus are equal in length to the lengths between the centers of the circular leaves.
The diagonals are formed by connecting opposite vertices of the rhombus.
The length of the diagonals can be found as follows:
- Diagonal 1: It connects the centers of two large circular leaves. The distance between their centers is 3 inches, as we determined earlier.
- Diagonal 2: It connects the centers of the other two large circular leaves. It is also 3 inches in length.
Therefore, the diagonals are both 3 inches long.
Now, let's calculate the area of the rhombus using the formula A = (d1 * d2) / 2:
A = (3 inches * 3 inches) / 2 = 9 square inches / 2 = 4.5 square inches
So, the area of the rhombus formed by the circular leaves is 4.5 square inches.