Chords, Secants, and Tangents

Lines and Circles

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Chords, secants, and tangents are basic geometric objects that describe how lines interact with circles. They appear in many geometry problems because they connect angle relationships, segment lengths, and properties of circles. Learning to identify each one helps students solve diagrams quickly and avoid mixing up similar terms. These ideas also show up in design, engineering, and any situation involving circular motion or curved boundaries.

A chord is a segment with both endpoints on a circle, a secant is a line that cuts through a circle at two points, and a tangent touches the circle at exactly one point. These definitions lead to useful theorems about lengths and angles. For example, a radius drawn to the point of tangency is perpendicular to the tangent line. Students often use power of a point relationships, such as tangent squared equals outside secant times whole secant, to solve for unknown lengths.

Key Facts

A chord is a line segment whose endpoints both lie on the circle.
A secant is a line that intersects a circle at two points.
A tangent is a line that touches a circle at exactly one point.
A radius to the point of tangency is perpendicular to the tangent line.
If two chords intersect inside a circle, then $a \cdot b = c \cdot d$ , where $a$ and $b$ are the parts of one chord and $c$ and $d$ are the parts of the other.
If a tangent and a secant are drawn from the same external point, then $t^2 = e \cdot (e + i)$ , where $t$ is the tangent length, $e$ is the external secant part, and $i$ is the internal secant part.

Vocabulary

Chord: A chord is a segment with both endpoints on a circle.
Secant: A secant is a line that passes through a circle and intersects it at two points.
Tangent: A tangent is a line that touches a circle at exactly one point.
Point of tangency: The point of tangency is the single point where a tangent touches a circle.
Radius: A radius is a segment from the center of a circle to a point on the circle.

Common Mistakes to Avoid

Calling any segment inside a circle a chord, which is wrong because a chord must have both endpoints on the circle.
Confusing a secant with a tangent, which is wrong because a secant crosses the circle at two points while a tangent touches it at only one point.
Forgetting that the radius to the point of tangency makes a right angle with the tangent, which is wrong because this perpendicular relationship is a key theorem used in many proofs and calculations.
Using the tangent-secant formula incorrectly, which is wrong because the correct relationship is tangent^2 = external part x whole secant, not external part x internal part.

Practice Questions

1 A secant from an external point has an outside segment of length 4 and an inside segment of length 9. Find the tangent length from the same external point.
2 Two chords intersect inside a circle. One chord is split into segments of lengths 3 and 8. The other chord is split into segments of lengths 4 and x. Find x.
3 Explain why a line through the endpoint of a radius is a tangent only when it is perpendicular to that radius.

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