OPQR is a rhombus whose 3 vertices P, Q,R lie on the circle with radius 8cm Find area of shaded region.
To find the area of the shaded region, we first need to find the area of the rhombus OPQR.
Since P, Q, and R lie on the circle with a radius of 8 cm, we can consider OP, OQ, and OR as radii of the circle. Therefore, OP = OQ = OR = 8 cm.
A rhombus is a special type of parallelogram where all sides are equal in length. In this case, all sides of the rhombus OPQR have a length of 8 cm.
The area of a rhombus can be calculated using the formula:
Area = (diagonal1 * diagonal2) / 2
Let's find the length of the diagonals in order to calculate the area:
1. The diagonals of a rhombus are perpendicular bisectors of each other. Therefore, the diagonal OR is also the diameter of the circle, and its length is twice the radius, which is 2 * 8 cm = 16 cm.
2. To find the length of the other diagonal, consider triangle OPR. The radii OP and OR are equal, so triangle OPR is an isosceles triangle.
Using the Pythagorean theorem, we can find the length of diagonal OQ:
(OQ)^2 = (OP)^2 + (PQ)^2
Since OP = OQ = 8 cm, we have:
(8 cm)^2 = (8 cm)^2 + (PQ)^2
Simplifying the equation:
64 cm^2 = 64 cm^2 + (PQ)^2
(PQ)^2 = 64 cm^2 - 64 cm^2
(PQ)^2 = 0 cm^2
PQ = 0 cm
Since PQ = 0 cm, the length of the other diagonal OQ is also 0 cm.
Now let's calculate the area of the rhombus using the formula:
Area = (diagonal1 * diagonal2) / 2
Area = (16 cm * 0 cm) / 2
Area = 0 cm^2
Therefore, the area of the shaded region is 0 cm^2.