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
Well, the area of the rhombus is half the product of the diagonals.
Draw a diagram and see where that takes you.
To find the radius of the circle, we need to first calculate the length of the diagonal of the rhombus.
A rhombus has all sides equal and diagonals that bisect each other at right angles. Let's label the vertices of the rhombus as A, B, C, and D, where AB = BC = CD = DA.
Since three of the vertices lie on the circle with center O, let's assume that these points are B, C, and D. Point B will be at the origin (0, 0).
Now, let's find the length of side BC using the area of the rhombus. The area of a rhombus is given by the formula:
Area = (d1 * d2) / 2
where d1 and d2 are the lengths of the diagonals.
In this case, we have:
Area = 32 * √3 cm²
Since the diagonals of a rhombus bisect each other at right angles, we can split the rhombus into four congruent right-angled triangles, each with a base and height that are half the lengths of the diagonals.
Let's assume that the length of BC is equal to x units. Therefore, the lengths of the diagonals will be 2x units.
The formula for the area of a triangle is given by:
Area of a triangle = (1/2) * base * height
In this case, the area of one of the right-angled triangles will be:
Area of one right-angled triangle = (1/2) * (x/2) * x = (1/4) * x²
Since there are four congruent triangles forming the rhombus, the total area of the rhombus is:
(1/4) * x² * 4 = x²
Now, we can set up the equation:
x² = 32 * √3
To solve for x, we take the square root of both sides:
x = √(32 * √3)
Now, since we need to find the radius, which is half the length of the diagonal, we have:
radius = x/2 = (√(32 * √3))/2
This will give us the final answer for the radius of the circle.