To find the distance between the two intersection points of two intersecting circles, we can use a formula derived from geometry. Let’s define the variables:

• Radius of the first circle ($r_1$) = 3
• Radius of the second circle ($r_2$) = 4
• Distance between the centers of the two circles ($d$) = 5

The distance between the two intersection points can be given by the following formula:

$d_{\text{intersect}} = 2 \times \sqrt{r_1^2 - \left( \frac{d^2 + r_1^2 - r_2^2}{2d} \right)^2}$

Step-by-step solution:

1. Calculate the term inside the parentheses:

   $\frac{d^2 + r_1^2 - r_2^2}{2d} = \frac{5^2 + 3^2 - 4^2}{2 \times 5} = \frac{25 + 9 - 16}{10} = \frac{18}{10} = 1.8$

2. Substitute this result into the main formula:

   $d_{\text{intersect}} = 2 \times \sqrt{3^2 - 1.8^2}$ $d_{\text{intersect}} = 2 \times \sqrt{9 - 3.24} = 2 \times \sqrt{5.76}$

3. Solve for $d_{\text{intersect}}$:

   $d_{\text{intersect}} = 2 \times 2.4 = 4.8$

So, the distance between the two intersection points is $\boxed{4.8}$ units.

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Would you like more details or have any questions? Here are some related questions you might be interested in:

1. What happens if the distance between the centers is less than $r_1 + r_2$?
2. How do we determine if two circles intersect or not?
3. What if the distance between the centers equals the sum of the radii?
4. How do you calculate the area of overlap between two intersecting circles?
5. How does the number of intersection points change if the radii or center distances change?

Tip: In geometry, when working with intersecting circles, always check if the circles actually intersect by comparing the distance between centers and the radii.