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.
Understanding Geometry: Chord Properties
Many chord rules come from a hidden pair of right triangles. Draw radii from the center to the two ends of a chord. Then draw a segment from the center that meets the chord at a right angle.
The two small triangles have equal radii as their hypotenuses, and they share the center segment. This makes the triangles congruent. Their matching pieces have the same length.
This reasoning matters more than memorising a rule because it shows why the midpoint appears. The reverse idea is useful too. A segment from the center to the midpoint of a chord must meet that chord at a right angle.
The distance between a chord and the center tells you how long the chord can be. A chord near the center is long. As it moves outward, it becomes shorter.
This can be calculated with the Pythagorean theorem. Half of the chord, the center-to-chord distance, and the radius form a right triangle. If a circle has radius ten units and the chord is six units from the center, half the chord has length eight units.
The full chord has length sixteen units. This method is often faster than trying to find the whole chord directly. A diameter is the special longest chord because its distance from the center is zero.
When two chords cross inside a circle, their four small pieces are linked by multiplication. The product of the two pieces on one chord equals the product of the two pieces on the other chord. This result comes from similar triangles formed around the crossing point.
It works even when the chords look very uneven. For example, one chord might be split into lengths three and eight. Its product is twenty-four.
If one piece of the other chord is six, its remaining piece must be four. Students should use the lengths from the crossing point, not the full chord lengths. Mixing up a segment with an entire chord is a common source of errors.
Chord properties connect closely to arcs and angles. Equal chords cut off equal arcs in the same circle. Larger chords cut off larger arcs, provided the arcs being compared are on the same side of the circle.
This helps when a diagram includes an inscribed angle, since an inscribed angle depends on the arc it intercepts. In real objects, chord measurements appear in circular windows, bridge arches, wheel parts, round tanks, and curved construction plans. Diagrams are often not drawn to scale, so visual guesses can mislead you.
Mark the center, right angles, equal radii, and known segment lengths before choosing a theorem. A clear labelled sketch usually reveals which relationship is available.
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.