Two equal circles intersect each other at points A and B, whose centres are O and O'. OO' = 24 mm and AB = 10 mm, then find the area (in sq. mm) of the circle.

Options

A. 530.66
B. 520.66
C. 510.33
D. 500.33

Answer

To find the area of the circle given that two equal circles intersect at points A and B, with centers O and O', we can follow these steps:

### Step 1: Understand the Geometry
The centers of the two circles, O and O', are 24 mm apart (OO' = 24 mm). The line segment AB, which connects the points of intersection of the circles, measures 10 mm.

### Step 2: Determine the Midpoint
Since the circles are equal and intersect symmetrically, we can find the midpoint M of segment AB. This midpoint divides AB into two equal parts:
- AM = MB = 10 mm / 2 = 5 mm.

### Step 3: Calculate Distances
The distance from the center O to the midpoint M (OM) can be found using the distance OO' and the properties of the right triangle formed by points O, M, and A.
- Since OO' = 24 mm, we can assume OM = 12 mm (half of OO').

### Step 4: Use the Pythagorean Theorem
In triangle OAM, we can apply the Pythagorean theorem:
- OA² = OM² + AM².

Substituting the known values:
- OM = 12 mm,
- AM = 5 mm.

Calculating:
- OA² = 12² + 5²,
- OA² = 144 + 25,
- OA² = 169.

### Step 5: Find the Radius
Now, take the square root to find OA (the radius of the circle):
- OA = √169 = 13 mm.

### Step 6: Calculate the Area of the Circle
The area A of a circle is given by the formula:
- A = πr².

Using π ≈ 3.14 and r = 13 mm:
- A = 3.14 × (13)²,
- A = 3.14 × 169,
- A ≈ 530.66 mm².

### Final Answer
The area of the circle is approximately **530.66 mm²**.

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