OPQR is a rhombus , three of whose vertices lie on a circle with centre O.If the area of the rhombus is 32 root 3cm square ,find the radius of the circle.
8cm
To find the radius of the circle, we need to first find the side length of the rhombus. Since a rhombus has all sides equal, we can use the formula for the area of a rhombus.
Given that the area of the rhombus is 32√3 cm², we can use the formula for the area of a rhombus:
Area = (1/2) × d₁ × d₂,
where d₁ and d₂ are the lengths of the diagonals of the rhombus.
Since the diagonals of a rhombus are perpendicular bisectors, if three vertices of the rhombus lie on a circle with center O, the diagonals of the rhombus are the diameters of the circle.
Let's assume the length of each side of the rhombus is "x".
The diagonals of the rhombus will be 2 times the radius of the circle (2r).
So, d₁ = 2r and d₂ = 2r.
Substituting these values into the area formula, we get:
32√3 = (1/2) × 2r × 2r.
Simplifying the equation, we get:
32√3 = 2r².
Dividing both sides by 2, we have:
16√3 = r².
To find the radius, we need to isolate "r". Taking the square root of both sides of the equation, we get:
r = √(16√3) = √(16) × √(√3) = 4√(√3) = 4√(3) cm.
Therefore, the radius of the circle is 4√3 cm.