There are number of line segments involved with a circle. The radius, diameter, tangent, chord and secant. Each one of them is defined by their special characteristics. In this article let us take a study on secants.

What is a Secant of a Circle?

As we mentioned earlier Secant of a circle is a line segment which passes through a circle. Since the secant is a straight line and circle is a curve, a secant intersects the circle only at two points. Let us study how it differs from other line segments like radius, diameter etc. in the following sub topic.

Secant of a Circle Definition

Let us take a look at the following diagram. The above diagram shows a circle with center at O and it also shows a number of line segments (as examples) associated with a circle. The description of each line segment is explained as follows.

OA - The line segment connecting the center of the circle and any point A on the circle is a radius of the circle.

BC - The line segment connecting any two points B and C on the circle which also passes through the center is a diameter of the circle.

ST - The line segment connecting any two points S and T on the circle but not passing through the center is a chord of the circle.

PQ - The line segment from any point P outside the circle and touching just touching any point Q on the circle is a tangent of the circle.

PT - The line segment from any point P outside the circle and correspondingly intersecting the circle at points S and T is a secant of the circle.

(The tangent and the secant can also be rays or lines).
Thus, the secant of a circle definition is a line segment (or a ray or a line) from a point outside the circle and passing through two points of the circle. If the points of intersection coincide, then the secant becomes as the tangent of the circle.

Theorems on Secant of a Circle

A circle can have any number of secants from a point outside the circle. The secants drawn to a circle from the same point outside the circle are called intersecting secants. There are a few theorems related to secants of a circle. Let us state and prove some of the important theorems on intersecting secants of a circle.
Theorem 1: The interior angle between two secants drawn outside a circle is half the difference of the measure of the arcs intercepted by the secants. (Secant angle theorem) In the above diagram, PAB and PCD are two intersecting secants from point P. As per the stated theorem,
measure of angle P = ($\frac{1}{2}$) (measure of arc BD - measure of arc AC)
For proving the theorem, let us join BC and AD. Let the intersect at E.

Considering triangles PBC and PDA,
angle PBC = angle PDA (angle subtended by the same arc AC on the circumference)
angle P is common.

Hence angle PCB = angle PAD (angle sum property)
The angle BAD = angle subtended by arc BD = ($\frac{1}{2}$) $\times$ measure of arc BD

Hence angle PAD = 180o - ($\frac{1}{2}$) $\times$ m of arc BD

So, angle PCB in triangle PCB also = 180o - ($\frac{1}{2}$) $\times$ measure of arc BD

Now in triangle PBC, angle PBC = angle subtended by arc AC = ($\frac{1}{2}$) $\times$ measure of arc AC
Since, measure of angle P + measure of angle PBC + measure of angle PCB = 180o,

measure of angle P + ($\frac{1}{2}$) $\times$ measure of arc AC + 180o - ($\frac{1}{2}$) $\times$ measure of arc BD = 180o
So, measure of angle P = ($\frac{1}{2}$) (measure of arc BD - measure of arc AC)

Theorem 2: When two secants of a circle intersect outside the circle, the product of the complete segment of the secant and the exterior segment of the secant is same for both the secants. (Intersecting secants theorem) Let us refer to the same diagram again. PB is a complete segment of secant PB and PA is the exterior segment of the same secant. In case of the secant PD, they are PD and PC respectively. As per this theorem, PA $\times$ PB = PC $\times$ PD

In the proof for the previous theorem we had shown that,
angle PBC = angle PDA (angle subtended by the same arc AC on the circumference)
angle P is common
angle PCB = angle PAD (angle sum property)
Therefore triangles PBC and PDA are similar.
Now as per the corresponding sides ratios of similar triangles,

$\frac{PB}{PD}$ = $\frac{PC}{PA}$

Therefore, PA $\times$ PB = PC $\times$ PD

Theorem 3: When a secant and a tangent of a circle intersect outside the circle, the product of the complete segment of the secant and the exterior segment of the secant is equal to the square of the segment of the tangent. (Secant - tangent intersection theorem)In fact, this theorem is a corollary of theorem 2. In the diagram shown above, if point B coincides with point A, then PB = PA. In other words, the secant PB becomes as a tangent PA. Therefore,
Therefore, PA $\times$ PB = PA2 = PC $\times$ PD
Thus theorem 3 is established.

Finding Secant of a Circle

For understanding the concept of secants of a circle, we will illustrate some example problems.

Solved Examples

Question 1: A circle is divided into six equal sectors as shown in the following diagram. Find the measure of the interior angle AEB. Solution:

Since all the sectors are equal in size, each sector has a central angle of 60o.
Hence, the measure of arc CD is 60o. AB is a diameter. Therefore angle AOB is
180o. In other words, the measure of arc APB = 180o.
Now looking at the diagram again, the line segments EA and EB are the intersecting secants of the circle and arcs CD and APB are the intercepted arcs by the secants.
Therefore, measure of angle AEB = ($\frac{1}{2}$) (measure of arc APB - measure of arc CD)
= ($\frac{1}{2}$)(180o – 60o) = ($\frac{1}{2}$) (120o) = 60o.
Thus the measure of internal angle AEB = 60o

Question 2: Find the value of ‘z’ in the following diagram. Solution:

In the above diagram ‘x’ is the angle by the arc AB on the circumference of the circle. Therefore, the arc AB would subtend an angle of ‘2x’ at the center of the circle. In other words, the measure of arc AB is 120o, since x = 60o.

Similarly, ‘y’ is the angle by the arc CD on the circumference of the circle. Therefore, the arc CD would subtend an angle of ‘2y’ at the center of the circle. In other words, the measure of arc CD is 50o, since x = 25o.

Now PA and PB are the secants and AB and CD are the intercepted arcs. Therefore as per secants-angle theorem,

z = ($\frac{1}{2}$) (measure of arc AB - measure of arc CD)

= ($\frac{1}{2}$)(120o – 50o) = ($\frac{1}{2}$) (70o) = 35o.

Thus the value of z = 35o.

Question 3: In the following diagram PB = 5 cm, BA = 4 cm and DC = 1.5 cm. Find the length of the secant PC. Solution:

In the above diagram PA and PC are two secants intersecting at P. Therefore, as per intersecting secants theorem,
PA* PB = PC* PD
(PB + BA) * (PB) = (PD + DC) * (PD)
(5 + 4) * (5) = (x + 1.5) * x, assuming PD = x
9 * 5 = x2 + 1.5x
45 = x2 + 1.5x
Multiplying by 2 throughout and rearranging,
2x2 + 3x – 90 = 0
or, (2x + 15)(x – 6) = 0
Ignoring the negative root, x = 6
Therefore, the length of the secant PC = x + 1.5 = 7.5 cm.

Question 4: A circular pulley is tied by a string to a peg along its circumference. The uncovered portion of the pulley subtends an angle of 120o at the center. The diameter of the pulley is 21 cm and the shortest distance between the pulley and the peg is 19 cm. Find the approximate length of the string. (Take the value of π as $\frac{22}{7}$)

The following diagram explains the situation describes the problem statement. Solution:

In the above diagram ACBDA is the pulley with center at 0. And ACPDA is the string. The portions PC and PD of the string act as tangents to the pulley and the rest of the string CAD rests on the circumference of the pulley. The interior angle COD is 120o and hence the exterior angle COD is 240o = ($\frac{4}{3}$)π radians.
The length of the portion CAD of the string = the length of arc CAD = r * exterior angle COD
= (2$\frac{1}{2}$) ($\frac{4}{3}$)π = (2$\frac{1}{2}$) ($\frac{4}{3}$)($\frac{22}{7}$) = 44 cm.
The line segment PBA is a secant of the circle and the line segment PC is the tangent of the circle, both intersecting at P. Therefore, as per secant-tangent intersection theorem,
PC2 = PB * PA = PB * (PB + BA) = (19) * (19 + 21) = 760
or, PC ≈ 27.57 cm
Similarly it can be shown that PD also ≈ 27.57 cm
Therefore, the approximate length of the string = arc length CAD + PC + PD
= 44 cm + 27.57 cm + 27.57 cm = 98.14 cm