The diagonals of a rhombus ABCD intersect at O. Taking 'O' as the centre, an arc of radius 6 cm is drawn intersecting OA and OD at E and F respectively. The area of the sector OEF is :
Get the complete, step-by-step math solution for: "The diagonals of a rhombus ABCD intersect at O. Taking 'O' as the centre, an arc of radius 6 cm is drawn intersecting OA and OD at E and F respectivel...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify properties of a rhombus
In a rhombus, the diagonals intersect at right angles. Therefore, the angle formed by the intersection of diagonals OA and OD, which is ∠AOD, is 90∘. This angle is also the central angle of the sector OEF.
Step 2: Determine the radius and central angle of the sector
The problem states that an arc of radius 6 cm is drawn with O as the center. This means the radius of the sector OEF is r=6 cm. From the previous step, we know the central angle θ=∠AOD=90∘.
Step 3: Apply the formula for the area of a sector
The formula for the area of a sector with radius r and central angle θ (in degrees) is given by 360∘θ×πr2. We will substitute the values of r and θ into this formula.
Step 4: Calculate the area of the sector
Substitute θ=90∘ and r=6 cm into the formula. Simplify the fraction 36090 to 41 and calculate 62=36.
Step 5: Final calculation
Multiply 41 by 36π to get the final area of the sector OEF.