What is the geometric relationship between a circle's center and the perpendicular bisector of any chord?

The perpendicular bisector of any chord always passes through the center of the circle. This occurs because the center is equidistant from the chord's endpoints, making it a point on the bisector.

How do you find the gradient of a line perpendicular to a chord with gradient $m$?

The perpendicular gradient is the negative reciprocal, calculated as $-\frac{1}{m}$. This ensures that the product of the two gradients equals $-1$, which is the condition for perpendicularity in a Cartesian plane.

When finding the equation of a circle from three points, why are two chords necessary?

One perpendicular bisector provides a line of possible center locations. A second bisector from a different chord provides another line; the unique point where these two lines intersect is the only possible center for the circle.

What is a common error when calculating the radius after finding the circle's center via chord bisection?

Students often mistakenly calculate the distance from the center to the chord's midpoint. The radius must be measured from the center to a point on the circumference (one of the chord's endpoints).

Define a 'chord' in the context of circle geometry.

A chord is a straight line segment whose endpoints both lie on the circumference of a circle. It divides the circle into two segments and its length is always less than or equal to the diameter.

How does the bisection of a chord relate to isosceles triangles?

If you connect the circle's center to the endpoints of a chord, you form an isosceles triangle because the two legs are radii. The perpendicular bisector of the chord acts as the altitude and median of this triangle.

What happens to the gradient calculation if a chord is perfectly horizontal?

A horizontal chord has a gradient of $0$. Its perpendicular bisector will be a vertical line with an undefined gradient, taking the form $x = k$, where $k$ is the x-coordinate of the chord's midpoint.

What is the difference between a chord's midpoint and the circle's center?

The midpoint is the center of the chord segment itself, while the circle's center is the point equidistant from all points on the circumference. The center only coincides with the midpoint if the chord is a diameter.

State the formula for the midpoint of a line segment.

The midpoint $M$ is calculated as $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$. This represents the average of the x-coordinates and the average of the y-coordinates of the segment's endpoints.

Why is the intersection of two perpendicular bisectors called the 'circumcenter'?

In the context of the triangle formed by three points on a circle, the circle is the 'circumcircle'. The center of this circle, found via chord bisection, is the circumcenter of that triangle.

Bisection of Chords | AQA A-Level Maths

Bisection of Chords

Summary

The bisection of chords is a fundamental geometric principle in circle geometry stating that the perpendicular bisector of any chord passes through the center of the circle. This relationship allows for the determination of a circle's center and equation using coordinate geometry techniques such as midpoints, gradients, and linear intersections.

1. Definition & Core Concepts

A chord is a straight line segment whose endpoints both lie on the circumference of a circle. Unlike a diameter, a chord does not necessarily pass through the center.
Bisection refers to the act of dividing a line segment into two equal parts. The point where this division occurs is the midpoint.
A perpendicular bisector is a line that intersects a chord at its midpoint at a $90^\circ$ angle. In circle geometry, this line is a line of symmetry for the chord relative to the circle's center.

Diagram showing a circle with a chord, its midpoint, and a perpendicular bisector passing through the center.

2. Underlying Principles

The Perpendicular Bisector Theorem for circles states that the perpendicular bisector of any chord is a diameter of the circle. Consequently, the center of the circle must lie somewhere along this line.
This principle is derived from the properties of isosceles triangles. If you draw radii from the center to the endpoints of a chord, you form an isosceles triangle where the perpendicular from the vertex (center) to the base (chord) bisects the base.
Because the center lies on the perpendicular bisector of any chord, the intersection of the perpendicular bisectors of two non-parallel chords uniquely identifies the center of the circle.

3. Methods & Techniques

4. Key Distinctions

Feature	Chord Bisector	Tangent Line
Intersection	Passes through the midpoint of a chord	Touches the circle at exactly one point
Relationship to Radius	Perpendicular to the chord at its midpoint	Perpendicular to the radius at the point of contact
Purpose	Used to find the circle's center	Used to find the circle's boundary or rate of change

It is vital to distinguish between the chord itself (a segment inside the circle) and the perpendicular bisector (a line that extends through the center).
While a diameter is a chord, not all chords are diameters; however, the perpendicular bisector of any chord will always contain the diameter.

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions