A secant is a straight line that cuts through a circle and touches it at exactly two points. In geometry, a circle is a round shape with parts like the center, radius, diameter, chord, tangent, secant, and arc. The secant is different from a chord (which lies only inside the circle) and a tangent (which touches the circle at just one point). While learning about secants, we also study their properties, how they compare with chords and tangents, and important theorems that explain how secants work. These theorems help solve problems related to circle geometry in an easy way.

What is Secant of a Circle?

A secant of a circle is a straight line that cuts through a circle and touches it at exactly two different points. These two points lie on the edge of the circle. The secant goes through the circle, entering one side and coming out the other. It passes through the circle’s inside, unlike a tangent which touches only one point. A straight line can cross a circle at most in two places, and when it does, it is called a secant. This concept is useful in geometry when studying how lines and circles interact with each other.

Example: Suppose that P and Q are any two points on the circumference of a circle and AB is the straight line and it intersects the circle at two points P and Q. So, the straight line AB is known as the secant of the circle at points P and Q (shown in the figure).

Properties of Secant of a Circle

The properties of the secant of a circle are as follows:

• Secant intersects the circle at two points.
• A secant is drawn from outside the circle.
• Secant is technically not a chord but it contains a chord.
• A tangent to a circle is a special case of the secant when two endpoints of its corresponding chord coincide.

Tangent and Secant of a Circle

A tangent is a line that touches the circle at only one point and a secant is a line that intersects the circle at two points.

A tangent to a circle is a special case of the secant when two endpoints of its corresponding chord coincide.

From the above figure, we can see that the secant line PQ becomes a tangent line as Q approaches P along the circumference of a circle or points P and Q coincide.

Tangent-Secant of a Circle Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.

In the circle, the CD is a tangent and AC is a secant.

So, \(CD^{2}=CA\times CB\)

Secant of a Circle Theorems

Intersecting Secants Theorem: If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.

In the circle, MO and MQ are secants that intersect at point M.

So, \(MN\times MO = MP\times MQ\).

Secant and Angle Measures: Two secants can intersect inside or outside the circle. There are two theorems based on secant and angle measures, which are as follows:

• If two secant lines intersect inside the circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. In the circle, the two secant lines AD and BC intersect inside the circle at point E. So, \(m\angle AEB = \frac{1}{2}(\overline{AB}+\overline{CD})\).

• If two secant lines intersect outside the circle, then the measure of an angle formed by the two lines is one-half the positive difference of the measures of the intercepted arcs.

In the circle, the two secant lines AC and AE intersect outside the circle at point A. So, \(m\angle CAE = \frac{1}{2}(\overline{CE}-\overline{BD})\).

Learn about Chord of a Circle

Properties of a Secant of a Circle

1. Intersects the Circle at Two Points
   A secant is a straight line that cuts through a circle, touching it at exactly two points.

2. Extends Beyond the Circle
   Unlike a chord, which stays within the boundary of the circle, a secant goes through the circle and extends on both sides.

3. Contains a Chord
   Every secant includes a chord—the segment between the two points where it touches the circle.

4. Used in the Intersecting Secants Theorem
   When two secants are drawn from a point outside the circle:

(Whole secant 1) × (External part of secant 1) = (Whole secant 2) × (External part of secant 2)

     5. Forms Angles with Tangents
A secant and a tangent from the same external point form specific angles and segment lengths, as explained by the secant-tangent theorem.

     6. Helps in Finding Lengths
Secants are useful in geometry problems involving lengths, angles, and the power of a point.

     7. Symmetry with Circle
If the secant passes through the center of the circle, it becomes a diameter—the longest chord—and divides the circle into two equal parts.

A chord is a line segment that links any two points on a circle. The two endpoints of a chord always lie on the circumference. In the following figure, the line segment AB is a chord.

A secant line is a line that goes through a circle and intersects the circle at two points. A secant is technically not a chord, but it contains a chord (the segment between the two red intersection points).

Important Notes About the Secant of a Circle

• A secant is a straight line that cuts through a circle and touches it at two different points.

• According to the Intersecting Secants Theorem:
  If you draw two secant lines from a point outside the circle, then:
  The length of one whole secant × its outer part = The length of the second whole secant × its outer part.

• According to the Secant-Tangent Rule:
  If you draw a secant line and a tangent line from the same point outside the circle, then:
  The length of the whole secant × its outer part = (Length of the tangent segment)²

Solved Examples of Secant of a Circle

Example 1: Two secants of a circle meet at a point outside the circle. One secant has 4 and ‘x’. The other secant is 5 and 4. What will be the ‘x’?

Solution: According to the question:

AB and CD intersect outside the circle at point P, such that

PB = x, AB = 4, PD = 4 AND CD = 5.

Using Intersecting Secant Theorem,

\(PA\times PB = PC\times PD\)

\((PB+AB)\times PB = (PD+CD)\times PD\)

\((x+4)\times x = (4+5)\times 4\)

\(x^{2}+4x=36\)

\(x^{2}+4x-36=0\), this is a quadratic equation in ‘x’.

By using the quadratic formula, we have

\(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)

\(x=\frac{-4\pm \sqrt{4^{2}-4(1)(-36)}}{2(1)}\)

\(x=\frac{-4\pm 4\sqrt{10}}{2}\)

\(x=-2(1 \mp \sqrt{10})\)

Hence, the values of x are \(x=-2(1-\sqrt{10})\) and \(x=-2(1+\sqrt{10})\).

Example 2: Which of the following is the secant line and chord to the circle:

1. AB and CD
2. CD and EF
3. EF and AB
4. None of the these

Solution: Let's understand the terms first:

• A chord is a line segment whose both endpoints lie on the circle.
• A secant is a line that cuts the circle at two points and extends beyond it.

Now, based on the given figure (assumed from the explanation):

• EF is a chord because both of its endpoints lie on the circle.
• CD is a secant because it intersects the circle at two points and extends outside.

Therefore, the correct answer is:

 Option (b) – CD and EF
Here, CD is the secant and EF is the chord.

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