We are tasked with finding the distance between the two intersection points of two circles, given the following details:

1. Circle 1 has a radius of $3$,
2. Circle 2 has a radius of $4$,
3. The centers of the circles are $5$ units apart.

Let me solve this step-by-step.

Step 1: Geometry of intersecting circles

When two circles intersect, the line segment connecting their centers ($d = 5$) and the line segment between the intersection points form a right triangle. The base of this triangle is the segment joining the centers of the circles, and the altitude is the line segment connecting the two intersection points.

The distance between the intersection points, which we denote as $L$, can be calculated using the following formula derived from geometry:

$L = 2 \sqrt{r_1^2 - \left(\frac{d^2 + r_1^2 - r_2^2}{2d}\right)^2}$

where:

• $r_1 = 3$ (radius of the smaller circle),
• $r_2 = 4$ (radius of the larger circle),
• $d = 5$ (distance between centers).

Step 2: Plugging values into the formula

First, calculate the term inside the parentheses:

$\frac{d^2 + r_1^2 - r_2^2}{2d} = \frac{5^2 + 3^2 - 4^2}{2 \cdot 5}$

$= \frac{25 + 9 - 16}{10} = \frac{18}{10} = 1.8$

Now, calculate $r_1^2 - \left(\frac{d^2 + r_1^2 - r_2^2}{2d}\right)^2$:

$r_1^2 - \left(\frac{d^2 + r_1^2 - r_2^2}{2d}\right)^2 = 3^2 - (1.8)^2$

$= 9 - 3.24 = 5.76$

Take the square root of $5.76$ and multiply by $2$:

$L = 2 \cdot \sqrt{5.76} = 2 \cdot 2.4 = 4.8$

Final Answer:

The distance between the two intersection points is 4.8 units.

Let me know if you'd like further clarifications!

Here are five related questions you might find interesting:

1. What happens if the distance between the centers is less than the sum of the radii but greater than the difference?
2. How can the formula for $L$ be derived geometrically?
3. What happens if the distance between the centers is exactly equal to the sum of the radii?
4. Can this problem be extended to three-dimensional spheres?
5. How can we find the area of the overlapping region between the two circles?

Tip: In geometry problems involving circles, sketching the situation often helps visualize the relationships between the quantities.