Area of the circle in which a chord of length`sqrt2` makes an angle `pi/2 `at the centre,

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `pi`
D. `2pi`

Answer

To find the area of the circle in which a chord of length \(\sqrt{2}\) makes an angle \(\frac{\pi}{2}\) (or 90 degrees) at the center, we can follow these steps:

### Step 1: Understand the Geometry
We have a circle with a chord \(AB\) of length \(\sqrt{2}\) that subtends an angle of \(90^\circ\) at the center \(O\) of the circle.

### Step 2: Draw the Triangle
Draw the triangle \(OAB\) where \(O\) is the center of the circle, and \(A\) and \(B\) are the endpoints of the chord. Since the angle \(AOB\) is \(90^\circ\), triangle \(OAB\) is a right triangle.

### Step 3: Apply the Pythagorean Theorem
In triangle \(OAB\):
- The length of the chord \(AB\) is given as \(\sqrt{2}\).
- The lengths \(OA\) and \(OB\) are both equal to the radius \(r\) of the circle.

Using the Pythagorean theorem:
\[ AB^2 = OA^2 + OB^2 \]
Substituting the known values:
\[ (\sqrt{2})^2 = r^2 + r^2 \]
\[ 2 = 2r^2 \]

### Step 4: Solve for the Radius
From the equation \(2 = 2r^2\):
\[ r^2 = 1 \]
Taking the square root:
\[ r = 1 \]

### Step 5: Calculate the Area of the Circle
The area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \(r\):
\[ A = \pi (1^2) = \pi \]

### Step 6: Approximate the Area
Using the approximation \(\pi \approx \frac{22}{7}\):
\[ A \approx \frac{22}{7} \text{ square units} \]

### Final Answer
The area of the circle is \(\frac{22}{7}\) square units.

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