Chords and arcs (theorems and relationships)
Chords and Arcs
Chords and Arcs is a topic in Circles in the California Common Core State Standards. It is aligned to Standard G-C.2, which requires students to identify and describe relationships among inscribed angles, radii, and chords.
Chords and arcs are linked: congruent chords cut congruent arcs, and a diameter perpendicular to a chord bisects it.
Theory
A chord is a segment joining two points on a circle. Several theorems connect chords, arcs, and the center:
- A radius perpendicular to a chord bisects the chord (and its arc).
- Congruent chords intercept congruent arcs.
- Chords equidistant from the center are congruent.
The right triangle from a chord:
How to use chord theorems
- Drop a perpendicular from the center to the chord; it bisects the chord.
- Form the right triangle with radius, distance, and half-chord.
- Apply the Pythagorean theorem to find the missing length.
The perpendicular from the center bisects the chord.
| \(\dfrac{16}{2}\) | \(=\) | \(8\) |
The radius, the distance, and half the chord form a right triangle.
| \(\text{half chord}\) | \(=\) | \(\sqrt{10^2-6^2}=\sqrt{64}=8\) |
| \(\text{chord}\) | \(=\) | \(2\times 8=16\) |
Congruent chords intercept congruent arcs.
Yes — chords equidistant from the center are congruent.
Common pitfalls
Frequently asked questions
What does a radius perpendicular to a chord do?
It bisects the chord and the arc the chord cuts off.
How do you find a chord's length from its distance to the center?
Use \(r^2=d^2+(\text{half chord})^2\), then double the half-chord.
What is true of congruent chords?
They intercept congruent arcs and are the same distance from the center.
Are chords equidistant from the center congruent?
Yes. Equal distance from the center means the chords are congruent.