To solve for the length of the common chord between two circles with radii 30 cm and 20 cm, and whose centers are 70 cm apart, we will apply the concept of the intersecting circles and use geometry to determine the length of the chord.

Given:

• Radius of the first circle, $r_1 = 30$ cm
• Radius of the second circle, $r_2 = 20$ cm
• Distance between the centers, $d = 70$ cm

Step 1: Formula for the length of the common chord

For two circles intersecting at two points, the length of the common chord can be calculated using the following formula:

$\text{Length of common chord} = 2 \sqrt{r_1^2 - d_1^2}$

where:

• $r_1$ is the radius of the first circle.
• $r_2$ is the radius of the second circle.
• $d$ is the distance between the centers of the circles.
• $d_1$ and $d_2$ are the perpendicular distances from the centers of each circle to the line joining the points of intersection.

Let's calculate the distance $d_1$ from the center of the first circle to the line joining the intersection points.

Step 2: Use the relationship between $d_1$ and $d_2$

We know that:

$d_1 = \frac{(r_1^2 - r_2^2 + d^2)}{2d}$

Plugging in the given values:

$d_1 = \frac{(30^2 - 20^2 + 70^2)}{2 \times 70}$

Let's simplify the equation:

$d_1 = \frac{(900 - 400 + 4900)}{140}$ $d_1 = \frac{5400}{140}$ $d_1 = 38.57 \, \text{cm}$

Step 3: Calculate the length of the common chord

Now we can use the formula for the length of the common chord:

$\text{Length of common chord} = 2 \sqrt{r_1^2 - d_1^2}$ $\text{Length of common chord} = 2 \sqrt{30^2 - 38.57^2}$ $\text{Length of common chord} = 2 \sqrt{900 - 1487.3}$

Since $d_1^2$ is greater than $r_1^2$, there is no real solution, meaning the circles do not intersect. The condition for intersection, i.e., the sum of the radii being greater than the distance between the centers and the difference of the radii being less than the distance between the centers, is not satisfied.

Thus, there is no common chord because the circles do not intersect.

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Would you like further clarification or details on the conditions for circle intersection? Here are some related questions that can expand your understanding:

1. How do we check if two circles intersect?
2. What happens when two circles are externally tangent to each other?
3. How do we find the area of intersection of two circles?
4. What are the properties of the common chord between two intersecting circles?
5. How do you calculate the distance between two tangent circles?

Tip: Always check if the circles intersect by comparing the distance between centers with the sum and difference of the radii!