Chords, Secants and Tangents
A chord joins two circle points, a secant line crosses the circle twice and a tangent line meets it at one point. In brief: a radius to a tangent point is perpendicular to the tangent, and two tangent segments drawn from the same external point have equal lengths. These facts connect circle geometry with right triangles and segment products.
The line types determine which theorem applies. Some results concern distances from the centre, while others relate segments from an intersection point. This guide develops chord properties, tangent lengths and secant products, with explicit attention to the difference between an external segment and the full length of a secant segment.
How do chords, secants and tangents differ?
A chord is a finite segment with both endpoints on a circle. A secant is a line that intersects the circle at two distinct points, and its interior portion is a chord. A tangent is a line meeting the circle at exactly one point, called the point of tangency.
A diameter is a chord through the centre and has length twice the radius. Other chords lie away from the centre and are shorter. A tangent has no interior chord because its two intersections have merged into one contact point in the limiting geometric picture.
The perpendicular distance from the centre to a line determines its type. A distance smaller than the radius gives a secant, equal to the radius gives a tangent and greater than the radius gives no intersection. This test is useful when a diagram alone does not make the relationship obvious.
How does a perpendicular from the centre relate to a chord?
The perpendicular from the centre of a circle to a chord bisects the chord. Joining the centre to both chord endpoints forms an isosceles triangle with equal radii, and the perpendicular to its base divides it into congruent right triangles. The chord’s two halves therefore have equal length.
Conversely, the line from the centre to the midpoint of a non-diameter chord is perpendicular to it. Equal chords in the same circle are equally distant from the centre, and chords closer to the centre are longer. These statements connect chord length with the right triangle formed by radius, centre distance and half-chord.
If a circle has radius 5 cm and a chord lies 3 cm from the centre, half the chord is √(5² − 3²) = 4 cm. The full chord length is 8 cm. Forgetting to double the right-triangle result gives only half the requested segment.
Why is a tangent perpendicular to the radius?
At the point of tangency T, the radius OT is perpendicular to the tangent line. The tangent’s distance from the centre equals the radius, and the perpendicular segment gives that shortest distance. A line through T that is not perpendicular to OT would enter the disk and meet the circle again.
This creates a right triangle when an external point P is joined to the centre O and the tangent point T. The hypotenuse is OP, one leg is radius r and the other is tangent length PT. Therefore PT² = OP² − r².
For OP = 13 cm and r = 5 cm, the tangent length is √(169 − 25) = 12 cm. The formula requires P to be outside the circle, so OP must exceed r for a positive tangent length. A point inside the disk has no real tangent segment to the circle.
Why are tangent segments from one external point equal?
If PA and PB are tangents from the same external point P, then PA = PB. The right triangles OAP and OBP share hypotenuse OP and have equal radius legs OA and OB. They are congruent, giving equal tangent lengths.
The phrase from the same external point is essential. Tangent segments drawn from different external points need not match. Likewise, the equal lengths are the finite segments from P to the contact points, not arbitrary lengths measured along the infinite tangent lines.
This theorem is useful in shapes that surround a circle. At each vertex of a tangential polygon, the two adjacent tangent pieces are equal. Labelling those repeated lengths can convert a perimeter problem into a sum of a few variables and reveal relationships among the side lengths.
What is the intersecting-chords theorem?
If chords AB and CD intersect at an interior point P, then PA × PB = PC × PD. Each product uses the two portions of one complete chord measured from the common intersection. The relationship follows from similar triangles formed using equal inscribed angles.
Suppose one chord is split into lengths 3 cm and 8 cm, while the other is split into 4 cm and x cm. Then 3 × 8 = 4x, so x = 6 cm. The products are square-length quantities even though the unknown result is a length.
Do not multiply adjacent pieces from different chords together. First identify which two segments belong to the same straight chord through P. A labelled sketch or a sentence naming the endpoints keeps the product pairs correct when several lines cross inside the circle.
| Configuration | Relationship |
|---|---|
| Chord half-length h, centre distance d, radius r | h² + d² = r² |
| External point distance D, tangent length t | t² = D² − r² |
| Intersecting chords at P | PA × PB = PC × PD |
| Two secants from P | External × whole is equal on both secants |
| Tangent and secant from P | Tangent² = external × whole |
How do you calculate lengths using two secants?
For two secants from one external point P, the product of the external part and the whole secant segment is the same on both lines. If the intersections are ordered P, A, B on one and P, C, D on the other, then PA × PB = PC × PD.
Suppose PA = 3 cm and AB = 5 cm, so the whole PB = 8 cm. If PC = 4 cm and CD = x cm, the other whole is 4 + x. The equation is 3 × 8 = 4(4 + x), giving 24 = 16 + 4x and x = 2 cm.
Using 3 × 5 would be incorrect because 5 is the internal chord portion, not the whole distance from P to B. This external-versus-whole distinction is the most important practical detail in secant calculations. Add the external and internal portions before forming the product when needed.
How does the tangent-secant theorem work?
For tangent PT and secant PAB from the same external point, PT² = PA × PB, where A is the nearer circle intersection and B the farther one. The tangent segment behaves as the limiting case of a secant whose two circle intersections coincide.
If the external secant part is 4 cm and the internal part is 5 cm, the whole secant is 9 cm. The tangent length satisfies t² = 4 × 9 = 36, so t = 6 cm. Choose the positive square root because the variable represents a physical segment length.
These product rules are related to the power of a point. For an external point at distance D from the centre, the positive power is D² − r², equal to both the tangent-length square and each external-times-whole secant product. The common value explains why different lines through the same point yield matching products.
How do you choose the correct chord, secant or tangent rule?
First locate the shared point: inside the circle, on it or outside it. Then identify whether the lines are chords, secants or tangents and label the required segments. An interior intersection suggests chord products, while an exterior point suggests secant products or a tangent square.
Check the geometric assumptions before calculating. A line is not a tangent merely because it appears to touch the sketch, and a right angle follows from a radius at the actual contact point. Use stated tangency or a proved perpendicular-distance condition rather than relying on the drawing’s appearance.
Finally, distinguish halves from full chords, external parts from whole secants and lengths from squared products. Substitute the result into the original relationship and verify that every segment length is positive and fits its containing segment. These checks catch many algebraically plausible answers that do not match the geometry.