Circle A has a center at #(3 ,4 )# and an area of #18 pi#. Circle B has a center at #(8 ,1 )# and an area of #40 pi#. Do the circles overlap?

To determine if the circles overlap, we need to compare the distances between the centers of the circles and the sum of their radii.

The distance between the centers of Circle A and Circle B can be calculated using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Next, we need to calculate the radii of both circles using the formula:

$Area = \pi r^2$

Once we have the radii, we can compare the distance between the centers of the circles with the sum of their radii.

If the distance between the centers is greater than the sum of the radii, the circles do not overlap. If it is equal to or less than the sum of the radii, the circles overlap.

To determine if the circles overlap, we can compare the distance between their centers to the sum of their radii. If the distance between the centers is less than the sum of their radii, the circles overlap. Otherwise, they do not overlap.

Let's denote the centers of the circles as $(x_1, y_1)$ for Circle A and $(x_2, y_2)$ for Circle B. The distance between the centers is given by the formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

And the radii of the circles can be calculated as follows:

For Circle A, radius $r_1 = \sqrt{\frac{A_1}{\pi}}$

For Circle B, radius $r_2 = \sqrt{\frac{A_2}{\pi}}$

Where $A_1$ and $A_2$ are the areas of Circle A and Circle B, respectively.

If $d < r_1 + r_2$, then the circles overlap.

Let's calculate:

For Circle A:

• Center: $(3, 4)$
• Radius: $r_1 = \sqrt{\frac{18 \pi}{\pi}} = \sqrt{18} = 3\sqrt{2}$

For Circle B:

• Center: $(8, 1)$
• Radius: $r_2 = \sqrt{\frac{40 \pi}{\pi}} = \sqrt{40} = 2\sqrt{10}$

Now, calculate the distance between the centers:

$d = \sqrt{(8 - 3)^2 + (1 - 4)^2} = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}$

So, $d = \sqrt{34}$.

Now, compare $d$ with $r_1 + r_2$:

$3\sqrt{2} + 2\sqrt{10} \approx 10.7$

Since $\sqrt{34} \approx 5.83 < 10.7$, the circles overlap.