Chord theorems translate curved information into straight-segment information. They also create right triangles that connect radius, half-chord, and distance from the center.
The chord theorem set
| Given in the same or congruent circles | Conclusion | Converse? |
|---|---|---|
| Congruent arcs | Their chords are congruent | Yes |
| Diameter perpendicular to a chord | It bisects the chord and intercepted arc | Use only the stated direction here |
| Chords equidistant from the center | The chords are congruent | Yes |
The chord-distance model
The perpendicular center-to-chord segment bisects the chord, so the triangle leg is c/2, not c.
Distance from center to chord
A circle has radius 13 and a chord of length 10. Find the perpendicular distance from the center to the chord.
- Bisect chord
Half the chord is 5.
- Pythagorean equation
\(d^2+5^2=13^2\), so \(d^2=144\).
Find a chord
A chord is 8 units from the center of a circle of radius 10. Find the chord length.
- Half-chord
\((c/2)^2+8^2=10^2\), so \(c/2=6\).
- Double
The complete chord is \(c=12\).
Chord check
A radius is 17 and the center-to-chord distance is 15. Find the chord.
Show answer and explanation
Answer: 16 units.
The half-chord is \(\sqrt{17^2-15^2}=8\), then double.
Key takeaways
- Congruent arcs and congruent chords correspond in the same or congruent circles.
- A perpendicular diameter bisects a chord and its arc.
- Equal center distances correspond to equal chord lengths.
- Use \(d^2+(c/2)^2=r^2\) and remember the half-chord.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.