Secant of a Circle

What is a Secant of a Circle?

Before learning what is a secant of a circle? We should first know what a circle is. A circle is the set of all those points in a plane, each of which is at a constant distance from a fixed point in the same plane. The fixed point of a circle is called the centre and the constant distance is called the radius. The radius of a circle is always positive and all radii of a circle are equal. 

Secant of a Circle Definition

Secant of a circle definition is as follows: A line that meets a circle in exactly two points. It is not a line segment and can be extended on both sides. Secant is different from chord, radius, diameter and tangent. As discussed above, the radius is the fixed distance from the centre to any point on the boundary of a circle. Diameter is double the radius. It is a line passing through the centre of a circle joining two points of the boundary of the circle. A chord is also a line segment joining any two points of the circle but unlike diameter, it doesn’t pass through the centre. Tangent touches only one point of the circle and is always perpendicular to one of the radii and diameters of the circle. The point where tangent meets the circle is called its point of contact. 

Difference Between Secant and Chord:

A secant intersects the circles at two points and the chord meets the circle at two points. Secant can be extended on both sides but chord cannot be extended. Parts of secant lie inside the circle and parts of secant lie outside the circle whereas chord lies inside the circle completely.

Difference Between Secant and Tangent: 

A secant intersects the circle at two points and tangent intersects the circles at one point. Both tangent and secant can be extended but parts of secant lie inside the circle and parts of secant lie outside the circle whereas tangent lies outside the circle completely.

Secant of a Circle Formula

Secant of a circle formula can be written as: Lengths of the secant × its external segment = (length of the tangent segment)2

The Theorem of Secants of a Circle

Case 1: Let us select an external point somewhere outside the circle. Now, if two secants are drawn from the external point such that each secant touches two points of the circle. In this case, each of the two lines can be divided into secant part and external part such that the whole part is a sum of both (i.e, a = b + secant and c = d + secant).        

Theorem 1:

The product of the first whole secant and its external part is equal to the product of the second whole secant and its external part. 

    a*b = c*d

Case 2: An external point is selected and a tangent and secant are drawn to a circle such that tangent touches one point and secant touch two points of the circle.

Theorem 2: The square of the length of the tangent is equal to the product of the whole secant and its external part. 

    a*a = c*b

FAQ (Frequently Asked Questions)

1. What is the Difference Between the Secant of a Circle and Secant Function?

Answer: In Latin, the word ‘secant’ is drawn from the word ‘secare’ that means to cut. It is a line that intersects two points on the curve. A chord is the interval of a secant lying between the points at which it intersects the curve. In a right-angled triangle, the secant of an angle will be:

The length of the hypotenuse will be divided by the length of the adjacent side.

The abbreviation is,

sec θ = hypotenuse / adjacent

2. What is Secant in a Circle? How is it Determined?

Answer: A secant in a circle is a line cutting through the circle at 2 points. In the limiting case, if the 2 points coincide, the secant becomes a tangent to the circle.

In order to find out whether a line is a secant to a circle, you need to ensure that the perpendicular distance from the centre to the line is less than or equal to the radius. therefore,  this would ensure that the line is cutting across the circle. If the distance equals to the radius, then, it’s the special case of a tangent.

x < R if the line is a secant

x = R for the tangent

3. Can a Diameter of a Circle Equal to the Secant to the Circle?

Answer: A secant of a curve is a line that intersects the curve at least twice.

In the case of a circle, a secant is a line that intersects the circle exactly two times. The line segment that joins the two points where the secant intersects the circle is called a chord.

If the secant passes through the centre of the circle, joining the two points of intersection is a diameter.

Therefore,  It can be seen that the diameter is a line segment whereas a secant is a line.

⇒A line segment can never be equal to the line on which it lies.

⇒ A diameter of a circle can never be equal to the secant of the circle.