Two circles, each with a radius of 20cm, are placed on a flat surface with their centres 32cm apart. Find a) the length of the common chord b) the area that is common to both circles.

Let's solve the problem step by step.

### Part a) Length of the Common Chord

Given:
- Radius of each circle, \( r = 20 \) cm
- Distance between the centers of the two circles, \( d = 32 \) cm

Let's denote the centers of the circles by \( O_1 \) and \( O_2 \). The common chord is perpendicular to the line segment \( O_1O_2 \) and bisects it.

To find the length of the common chord, we can use the following formula:

\[
\text{Length of the common chord} = 2 \sqrt{r^2 - \left(\frac{d}{2}\right)^2}
\]

Substituting the values:

\[
\text{Length of the common chord} = 2 \sqrt{20^2 - \left(\frac{32}{2}\right)^2}
\]

\[
\text{Length of the common chord} = 2 \sqrt{400 - 256}
\]

\[
\text{Length of the common chord} = 2 \sqrt{144}
\]

\[
\text{Length of the common chord} = 2 \times 12 = 24 \text{ cm}
\]

### Part b) Area Common to Both Circles

The area common to both circles can be found by calculating the area of each circular segment and then doubling it (since the two segments are symmetrical).

To find the area of one segment, we use the formula:

\[
\text{Area of a segment} = r^2 \cos^{-1}\left(\frac{d}{2r}\right) - \frac{d}{2} \sqrt{r^2 - \left(\frac{d}{2}\right)^2}
\]

Here \( \cos^{-1}\left(\frac{d}{2r}\right) \) is the angle subtended by the chord at the center of the circle.

Substituting the values:

\[
\text{Area of one segment} = 20^2 \cos^{-1}\left(\frac{32}{2 \times 20}\right) - \frac{32}{2} \times \sqrt{20^2 - \left(\frac{32}{2}\right)^2}
\]

Let's calculate the area of one segment.

\[
\cos^{-1}\left(\frac{32}{40}\right) = \cos^{-1}\left(\frac{4}{5}\right) \approx 0.6435 \text{ radians}
\]

\[
\text{Area of one segment} = 400 \times 0.6435 - 16 \times 12
\]

\[
\text{Area of one segment} \approx 257.4 - 192 = 65.4 \text{ cm}^2
\]

Since the common area is twice the area of one segment:

\[
\text{Common area} \approx 2 \times 65.4 = 130.8 \text{ cm}^2
\]

### Final Answers:
a) The length of the common chord is **24 cm**.  
b) The area that is common to both circles is approximately **130.8 cm²**.

Would you like more details or have any questions about this problem?

### Related Questions:
1. How does the distance between the centers affect the length of the common chord?
2. What is the area of one of the circles?
3. How would the common area change if the radius of the circles increased?
4. What if the circles did not intersect; how would we calculate the distance between them?
5. How is the angle subtended by the chord at the center calculated?
6. Can we find the common area using a different method?
7. What if the distance between the centers is less than the radius?
8. How do these calculations change if the circles are not on a flat surface?

### Tip:
When solving geometry problems involving circles, sketching the scenario can help visualize the relationships between different elements.

Learn how to solve the geometry problem involving two circles with radius 20 cm each, placed 32 cm apart. Find the length of the common chord and the area that is common to both circles using step-by-step calculations and circle geometry formulas.

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