We are given two intersecting circles:

• Circle 1 has a radius of 3.
• Circle 2 has a radius of 4.
• The distance between their centers is 5.

The question asks us to find the distance between the two points where these circles intersect.

To solve this, we can use geometric properties of intersecting circles. Let’s consider this configuration geometrically:

1. The two circles intersect at two points, so we form a triangle with:

   • One vertex at the center of the first circle.
   • One vertex at the center of the second circle.
   • The third vertex is any one of the two intersection points.

2. Let the distance between the centers of the two circles be denoted as $d = 5$. The line connecting the centers is perpendicular to the line connecting the two intersection points, and the distance between these points is what we need to calculate.

To calculate the distance between the intersection points, we use the following formula for the distance between the intersection points of two circles: $\text{distance between intersection points} = 2 \sqrt{r_1^2 - \left( \frac{d^2 + r_1^2 - r_2^2}{2d} \right)^2}$ Where:

• $r_1 = 3$ (radius of the first circle)
• $r_2 = 4$ (radius of the second circle)
• $d = 5$ (distance between the centers)

Let me calculate this for you.The distance between the two intersection points of the circles is approximately 4.8 units.

Would you like more details on this solution or have any further questions?

Here are 5 related questions to expand your understanding:

1. How is the formula for the distance between intersection points derived?
2. What happens if the two circles don't intersect?
3. Can this method be used for any pair of intersecting circles, regardless of their radii?
4. How does the distance between the centers affect the intersection points?
5. How would the solution change if the radii of the circles were larger than the distance between their centers?

Tip: When dealing with geometric problems, visualizing the situation can often provide insight into how to approach the problem. Try sketching the circles and their intersection points!