To solve the problem of finding the distance between the two intersection points of two circles, we can use the following approach:

Step 1: Understand the geometry

You have two intersecting circles:

• Circle 1: Radius $r_1 = 3$
• Circle 2: Radius $r_2 = 4$
• Distance between the centers $d = 5$

We need to find the distance between the two intersection points of these circles.

Step 2: Use the formula for the distance between intersection points

The distance between the two intersection points of two intersecting circles can be found using the following formula:

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

Where:

• $r_1$ and $r_2$ are the radii of the circles
• $d$ is the distance between the centers of the two circles

Step 3: Plug in the values

Substitute the given values into the formula:

$\text{Distance} = 2 \times \sqrt{3^2 - \left(\frac{5^2 + 3^2 - 4^2}{2 \times 5}\right)^2}$

First, calculate the term inside the square root:

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

Now, calculate the square of this value:

$1.8^2 = 3.24$

Subtract this from $r_1^2$:

$3^2 - 3.24 = 9 - 3.24 = 5.76$

Finally, find the square root and multiply by 2:

$\text{Distance} = 2 \times \sqrt{5.76} = 2 \times 2.4 = 4.8$

So, the distance between the two intersection points of the circles is 4.8 units.

Would you like more details or have any questions?

Here are five related questions:

1. How would the formula change if the circles did not intersect?
2. Can you derive this formula using the properties of triangles?
3. What happens to the distance if the radii of the circles are equal?
4. How does the position of the intersection points change if the distance $d$ increases?
5. What would the scenario look like if $d = r_1 + r_2$?

Tip: For geometric problems involving circles, visualizing the problem with a diagram often helps in understanding the relationships between different elements.