The center of a circle is at (-3,2)) and its radius is 7. Find the length of the chord, which is bisected at (3, 1).

To find the length of the chord that is bisected at the point \((3, 1)\) in a circle with center \((-3, 2)\) and radius \(7\), we can follow these steps:

### Step 1: Calculate the distance from the center to the midpoint of the chord

The midpoint of the chord is given as \((3, 1)\). The center of the circle is \((-3, 2)\). To find the distance \(d\) between these two points, we use the distance formula:

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

Substituting the given points:

\[
d = \sqrt{(3 - (-3))^2 + (1 - 2)^2}
\]
\[
d = \sqrt{(3 + 3)^2 + (1 - 2)^2}
\]
\[
d = \sqrt{6^2 + (-1)^2}
\]
\[
d = \sqrt{36 + 1}
\]
\[
d = \sqrt{37}
\]

### Step 2: Use the perpendicular bisector property of the chord

In a circle, the line segment joining the center of the circle to the midpoint of the chord is perpendicular to the chord. Thus, we can consider a right triangle where:

- One leg is the distance from the center to the midpoint of the chord (\(\sqrt{37}\)).
- The other leg is half the length of the chord (let's call it \(\frac{L}{2}\)).
- The hypotenuse is the radius of the circle (\(7\)).

Using the Pythagorean theorem:

\[
\left(\frac{L}{2}\right)^2 + \left(\sqrt{37}\right)^2 = 7^2
\]

\[
\frac{L^2}{4} + 37 = 49
\]

### Step 3: Solve for \(L\)

\[
\frac{L^2}{4} = 49 - 37
\]

\[
\frac{L^2}{4} = 12
\]

\[
L^2 = 48
\]

\[
L = \sqrt{48}
\]

\[
L = \sqrt{16 \cdot 3}
\]

\[
L = 4\sqrt{3}
\]

Therefore, the length of the chord is \(4\sqrt{3}\) units.

### Summary

The length of the chord that is bisected at the point \((3, 1)\) in the given circle is \(4\sqrt{3}\) units.

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Would you like more details on any of these steps, or do you have any other questions?

### Related Questions:

1. How do you find the equation of the circle given its center and radius?
2. How can you verify if a point lies on the circumference of a circle?
3. What is the formula for the length of a chord given its distance from the center of the circle?
4. How can you find the coordinates of the endpoints of the chord given its midpoint and length?
5. How would the solution change if the radius or the center of the circle were different?

### Tip

Always use the distance formula to find the straight-line distance between two points, which is critical in solving geometry problems involving circles and other shapes.

Learn how to find the length of a chord bisected at the point (3, 1) in a circle with center (-3, 2) and radius 7 using circle geometry principles. Step-by-step solution using the distance formula, perpendicular bisector theorem, and Pythagorean theorem provided.

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