A chord is a line segment whose endpoints lie on a circle, and chord properties help connect geometry diagrams to exact measurements. Chords appear in many circle problems because they create triangles, distances from the center, arcs, and angles. Learning these properties makes it easier to solve for missing lengths without guessing from a drawing.
They are also useful in design, surveying, engineering, and any situation involving circular shapes.
Key Facts
- A chord is any segment with both endpoints on the circle.
- The perpendicular from the center of a circle to a chord bisects the chord.
- If OM is perpendicular to chord AB, then AM = MB.
- Equal chords in the same circle are equidistant from the center.
- Chords that are equidistant from the center of the same circle are equal in length.
- For intersecting chords, if chords AB and CD intersect at P, then AP × PB = CP × PD.
Vocabulary
- Chord
- A chord is a line segment whose endpoints both lie on the circle.
- Diameter
- A diameter is a chord that passes through the center of the circle.
- Radius
- A radius is a segment from the center of a circle to any point on the circle.
- Perpendicular bisector
- A perpendicular bisector is a line or segment that crosses another segment at a right angle and divides it into two equal parts.
- Intersecting chords theorem
- The intersecting chords theorem states that the products of the two segment lengths on each chord are equal when two chords intersect inside a circle.
Common Mistakes to Avoid
- Assuming every line through a circle is a chord. A chord must have both endpoints on the circle, while a secant line continues beyond the circle.
- Forgetting that the center-to-chord segment must be perpendicular before using bisection. The center only bisects the chord when the segment from the center meets the chord at a right angle.
- Treating equal-looking chords as equal without proof. Chords are equal only if given equal lengths, shown congruent, or proven equidistant from the center in the same circle.
- Adding intersecting chord segments instead of multiplying them. The theorem uses products, so AP × PB = CP × PD, not AP + PB = CP + PD.
Practice Questions
- 1 In a circle with center O, chord AB is 16 cm long. Segment OM is perpendicular to AB at M. What are AM and MB?
- 2 Two chords intersect at P. On one chord, AP = 6 and PB = 10. On the other chord, CP = 5. Find PD.
- 3 In the same circle, chord AB and chord CD are the same distance from the center. Explain what must be true about AB and CD, and state the chord property that justifies your answer.