Question

Circles A and B, both of which have a radius of 4 inches, pass through each other's centers. What is the distance from point C to point D in inches?

Answer 1

Let the centers of circles A and B be points O_A and O_B, respectively. Since the radius of each circle is 4 inches and each circle passes through the center of the other, the distance between O_A and O_B is equal to the radius of one circle, which is 4 inches.

Next, we can place the circles on a coordinate plane for convenience:

- Let the center of circle A, O_A, be at the origin, (0, 0).
- The center of circle B, O_B, is then at (4, 0) because it is 4 inches to the right of O_A.

Since both circles have a radius of 4 inches, we can identify the points on the circumference where circles A and B intersect.

1. For circle A (centered at O_A = (0, 0)), the equation is given by:
\[
x^2 + y^2 = 4^2 = 16.
\]

2. For circle B (centered at O_B = (4, 0)), the equation is given by:
\[
(x - 4)^2 + y^2 = 4^2 = 16.
\]

We can solve for the points of intersection by substituting the coordinates from one equation into the other.

### Step 1: Using circle A's equation

We'll express y from circle A's equation:
\[
y^2 = 16 - x^2.
\]

### Step 2: Substitute into circle B's equation

Substituting \( y^2 \) into circle B's equation gives:
\[
(x - 4)^2 + (16 - x^2) = 16.
\]
Expanding and simplifying:
\[
(x - 4)^2 + 16 - x^2 = 16,
\]
\[
(x - 4)^2 - x^2 = 0.
\]
Expanding \((x - 4)^2\):
\[
x^2 - 8x + 16 - x^2 = 0,
\]
which simplifies to:
\[
-8x + 16 = 0.
\]
Solving for \( x \):
\[
8x = 16 \implies x = 2.
\]

### Step 3: Finding y

Substituting \( x = 2 \) back into the equation for circle A to find y:
\[
y^2 = 16 - 2^2 = 16 - 4 = 12 \implies y = \pm 2\sqrt{3}.
\]

Thus, the points of intersection are:
\[
(2, 2\sqrt{3}) \quad \text{and} \quad (2, -2\sqrt{3}).
\]

### Step 4: Finding the distance from points C to D

Let point C be (2, 2√3) and point D be (2, -2√3). The distance \( d \) between these two points is calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
\]
Substituting the coordinates:
\[
d = \sqrt{(2 - 2)^2 + (-2\sqrt{3} - 2\sqrt{3})^2} = \sqrt{0 + (-4\sqrt{3})^2} = \sqrt{16 \cdot 3} = \sqrt{48} = 4\sqrt{3}.
\]

Therefore, the distance from point C to point D is:
\[
\boxed{4\sqrt{3}} \text{ inches.}
\]