Key takeaway: A chord is a line segment whose two endpoints both lie on a circle, and the longest chord through any circle is a diameter that passes through the center.
Grade 10 – GeometryG-C.A.2Geometry and Measurement
Definition of Chord
A chord is a straight segment drawn from one point of a circle to another. Unlike a secant, which is a full line, it stops at the circle. A diameter is the special case passing through the center, which makes it the longest segment of this kind.
These segments obey tidy rules. A radius drawn perpendicular to one of them cuts it into two equal halves, so the center always lies on its perpendicular bisector. Two segments of the same length sit equally far from the center, and the longer a segment is, the closer it lies to the center.
Formula and notation
Name it by its endpoints, \(\overline{AB}\). If \(d\) is the distance from the center and \(r\) is the radius, the length is:
\[ AB = 2\sqrt{r^2 – d^2} \]
This comes from the right triangle made by the radius, the distance \(d\) and half of the segment.
Key facts
- A diameter passes through the center and equals twice the radius.
- A perpendicular dropped from the center bisects the segment.
- Equal-length segments are the same distance from the center.
- Equal lengths also cut off equal curved pieces of the circle.
Chord example
A circular fountain has a radius of 17 ft. A straight pipe runs along a chord that is 8 ft from the center. How long is the pipe?
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The perpendicular from the center bisects the pipe, forming a right triangle with hypotenuse 17 and one leg 8.
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Half of the pipe: \(\sqrt{17^2 – 8^2} = \sqrt{289 – 64} = \sqrt{225} = 15\) ft.
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Double it for the whole pipe: \(2 \times 15 = 30\) ft.
The pipe is 30 ft long.
Common mistake
Watch out
Imani answers 15 ft, the result of the square root, as the length of the pipe.
Why it is wrong
The square root gives only half of the pipe, from the midpoint to one end. The whole chord is double that, 30 ft. A check: nothing in this circle can exceed the 34-ft diameter, and 30 ft passes.
Related terms and lessons
- Arc (Circle)
- Central Angle
- Inscribed Angle
- Perpendicular Bisector
- Radius and Diameter
- Pythagorean Theorem
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