Secant of a Circle

A secant of a circle is a line that intersects a circle at two distinct points. Secant is derived from the Latin word secare which means to cut. It can also be understood as the extension of the chord of a circle that goes outside the circle.

Secant of a circle is the line that cuts across the circle intersecting the circle at 2 distinct points. In the circle below, PQ is the secant line that cuts the circle at two points A and B.

Difference Between a Chord and a Secant

When a secant line cuts the circle at two points, we get a chord at the two points of intersection. The chord of a circle is a line segment whose endpoints lie on the circular arc. In the circle shown above, AB is the chord which is a portion of the secant line QP. In other words, a chord is a line segment joining two points on the circumference of the circle, and if this chord is extended on both sides it becomes the secant. The secant line that passes through the center of the circle produces the diameter. Thus a secant line determines the chord or diameter in a circle.

In real life, we come across a secant of a circle in many places, wherever the circles or curves are involved. For example, in the construction of curved bridges, in finding the distance between the orbiting moon and the different locations on earth, and so on. There are many interesting properties of secants that help in obscure geometric constructions. There are many circle theorems based on the secants and the intersecting secants of a circle.

The intersecting secants theorem states that when two secants intersect at an exterior point, the product of the one whole secant segment and its external segment is equal to the product of the other whole secant segment and its external segment. This is also known as the secant theorem or the secant power theorem.

In the figure shown above, we find that AB and AC are the two secant segments intersecting at point A. AD is the external secant segment of the whole secant segment AB, and AE is the external secant segment of AC. Thus, according to the theorem, we have AB × AD = AC × AE

Secants and Angle Measures

Two secants can intersect inside or outside a circle. In the circles shown below, we find that the intersecting secants inside and outside create angles x and y at the points of intersection, respectively. In the first circle, the secants intersect inside the circle and major arc AD and minor arc BD are intercepted by the secants. In the second circle, the secants intersect outside the circle and the major arc PT and minor arc QS are intercepted by the secants.

There are two theorems based on this property of secants. As per the theorem, we have:

• The angle formed by the two secants that intersect inside the circle is half the sum of the intercepted arcs.
• The angle formed by the two secants which intersect outside the circle is half the difference of the intercepted arcs.

Tangents and secants are the lines that cut the circle and extend in both directions infinitely. The main difference between them is that a secant cuts the circle at two points, whereas, a tangent cuts the circle at one point. The tangent is perpendicular to the radius at the point of the tangency.

Tangent Secant Theorem

According to the tangent secant theorem, if a secant and a tangent are drawn to a circle from a common exterior point, then the product of the length of the whole secant segment and its external secant segment is equal to the square of the length of the tangent segment.

Observe the figure given above to see that:

• The secant AC and the tangent CD are drawn from the same exterior point. Secant segments are AB (interior) and BC(exterior). The product of the secant and its exterior segment is equal to the square of the tangent segment. AC × BC = CD²
• The angle subtended by the tangent and the secant at the exterior is half the difference of the major arc and the minor arc intercepted by them. \(\alpha = \dfrac{1}{2}[ \overline{\rm AD} - \overline{\rm BD}]\)

Check out here for a few interesting topics related to secant of a circle:

• Slope of the Secant Line Formula
• Segment of a Circle
• All Circle Formulas
• Circumference of a Circle

Important Notes

Here is a list of a few points that should be remembered while studying about the secant of a circle:

• A secant of a circle is a line that connects two distinct points on a curve.
• The intersecting secants theorem states that if we draw two secant lines from an exterior point of a circle, the product of one secant and its external segment is equal to the product of the other secant and its external segment.
• The secant-tangent rule states that when a secant line and a tangent line are drawn both from a common exterior point, the product of the secant and its external segment is equal to the square of the tangent segment.

What is the Secant of a Circle?

A line that intersects the circle exactly at two distinct points is the secant of the circle. A secant line includes the chord and is always extended outside the circle. In other words, if the chord is extended on both sides it becomes the secant.

Is Secant of a Circle and the Chord of a Circle the Same?

No, a secant and a chord are different. A chord is a portion of the secant with its endpoint on the secant line. A secant is obtained by extending the chord infinitely in both directions. A chord lies within the circle, whereas, a secant extends outside the circle.

What is the Tangent Secant Theorem?

According to the tangent secant theorem, when a secant and a tangent are drawn to a circle and if they intersect at an exterior point on the circle, they are associated in such a way the square of the tangent is equal to the product of the whole secant segment and the exterior secant segment.

What is the Intersecting Secants Theorem?

According to the Intersecting secants theorem, when two secants intersect at an exterior point, the product of the one whole secant segment and its external segment is equal to the product of the other whole secant segment and its external segment.

Is a Secant of a Circle Always a Chord?

A secant can never be a chord, but it contains the chord. An extension of the chord on both sides is the secant of the circle.

How many Secants Can a Circle Have?

There can be an infinite number of secants drawn to a circle. Every secant intersects the circle at two points.