Tangent Secant Theorem

Tangent Secant Theorem is the fundamental theorem in geometry. Tangent and secant are the important parts of the circle. The tangent secant theorem is used in various fields of mathematics, construction, and many more. Tangents and secants are the lines that intersect the circle at some points.

In this article, we will learn about the Tangent Secant theorem in detail along with its statement and proof. It also covers the applications and limitations of the tangent secant theorem and some solved examples of the Tangent Secant Theorem. Let’s start our learning on the topic Tangent Secant theorem.

What is Tangent and Secant?

Tangent and Secant are line segments or lines related to a curve, which help us understand its behaviour and characteristics at specific points and between multiple points along the curve. In simple words, any line that touches the curve at only one point is called a tangent, while a line that intersects the curve at two points is called a secant.

What is Tangent to a Circle?

Tangent to a circle is a straight line that touches the circle at exactly one point without intersecting it. This point of contact is called Point of Tangency.

Tangent line to a circle is always perpendicular to the radius of the circle at the point of tangency i.e., the radius and the tangent line form a right angle at the point where they meet. In the above figure AB is the tangent of the circle.

What is Secant to a Circle?

Secant to a circle is also a straight line similar to tangent, however this line intersects the circle at exactly two distinct points. In simple words, secant is a line which cuts through the circle and pass through it’s interior. In the above figure ACD is the secant of the circle.

Read more about Circle.

What is Tangent Secant Theorem?

The tangent secant theorem as the name suggests states the geometric relationship between the lengths of tangent and secant of any circle. Tangent-Secant Theorem is also known as the Secant-Tangent Theorem. We will discuss the statement of tangent secant theorem below.

Tangent Secant Theorem Statement

The tangent secant theorem states that when a tangent and secant drawn from the point outside the circle then, the square of length of the tangent is equal to the total length of secant multiplied by the outer length of the secant.

In the above figure O is the center of the circle, AB be the tangent of the circle from the external point A and ACD be the secant of the circle where C and D are the points on the circle.

According to the Tangent Secant Theorem

AB² = AD × AC

Proof of Tangent Secant Theorem

Consider the figure below, where O is the center of the circle ACD is secant of the circle and AB be the tangent on the circle. A line OP is drawn perpendicular to CD. Join OC, OA and OB.

Now, since OP ⟂ CD

CP = PD —(1)

[Perpendicular drawn from the center of the circle on the chord bisects the chord]

AC × AD = (AP – CP) (AP + PD)

⇒ AC × AD = (AP – CP) (AP + CP) [From 1]

⇒ AC × AD = AP² – CP²

In △ OAP

OA² = OP² + AP²

⇒ AP² = OA² – OP²

⇒ AC × AD = AP² – CP²

⇒ AC × AD = OA² – OP² – CP²

⇒ AC × AD = OA² – (OP² + CP²)

In △ OCP

OC² = OP² + CP²

⇒ CP² = OC² – OP²

⇒ AC × AD = OA² – (OP² + CP²)

⇒ AC × AD = OA² – (OP² + OC² – OP²)

⇒ AC × AD = OA² – OC²

Since OC = OB

Thus, AC × AD = OA² – OB²

In △ OAB

OA² = OB² + AB²

⇒ AB² = OA² – OB²

⇒ AC × AD = OA² – OB²

⇒ AC × AD = AB²

Hence proved

Limitation and Applications of Tangent Secant Theorem

The tangent secant theorem has both applications and limitations. Below we will discuss the limitations and applications of the tangent secant theorem in detail.

Limitation of Tangent Secant Theorem

Along with the applications the tangent secant theorem has some limitations. Some of the limitations of the tangent secant theorem are listed below:

• Tangent Secant Theorem is not applicable to three-dimensional shapes.
• The tangent secant theorem does not provide any information about whether the secant and tangent are drawn from the same points.

Applications of Tangent Secant Theorem

The tangent secant theorem has multiple applications in real life. Some of these applications are listed below:

• The construction of buildings and bridges is based on the tangent secant theorem.
• The construction of statutes and pyramids are also based on the tangent secant theorem.

Some More Theorem in Geometry

• Pythagoras Theorem
• Basic Proportionality Theorem
• Mid Point Theorem

Solved Problems on Tangent Secant Theorem

Example 1: Find the value of x.

Solution:

Total length of the secant = 6 + 10 = 16

By the tangent secant theorem

AB² = AC × AD

⇒ x² = 6 × 16

⇒ x² = 96

⇒ x = 4√6

Example 2: Find the total length of the secant.

Solution:

By the tangent secant theorem

AB² = AC × AD

⇒ 10² = 6 × (x + 2)

⇒ (x + 2) = 100 / 6

⇒ (x + 2) = 16.66

Total length of the secant CD = x + 2 = 16.66 = 14.66

Example 3: Find the length of the tangent.

Solution:

By the tangent secant theorem

AB² = AC × AD

⇒ z² = 8 × 15

⇒ z² = 120

⇒ z = 2√30

Example 4: Find the value of x.

Solution:

By the tangent secant theorem

AB² = AC × AD

⇒ 12² = x × (x + 5)

⇒ 144 = x² + 5x

⇒ x² + 5x – 144= 0

⇒ x = 9.75 or x = -14.75 (length cannot be negative)

Thus, x = 9.75

Example 5: Find the value of x and y.

Solution:

By the tangent secant theorem

AB² = AC × AD

⇒ 9² = x × 12

⇒ x = 81 / 12

⇒ x = 6.75

From the above figure

x + y = 12

⇒ y = 12 – x

⇒ y = 12 – 6.75

⇒ y = 5.25

Practice Problems on Tangent Secant Theorem

Problem 1: In a circle with a radius of 5 cm, point P is located 13 cm away from the center O. A tangent is drawn from point P to the circle, and it touches the circle at point T. Calculate the length of PT.

Problem 2: In a circle with a radius of 8 cm, a secant is drawn from an external point P. The external portion of the secant is 12 cm, and the entire secant is 16 cm long. Find the length of the tangent segment from point P to the circle.

Problem 3: In a circle with a radius of 6 cm, a secant line is drawn from an external point P such that the tangent segment formed from P to the circle is 8 cm long. Find the length of the entire secant.

Problem 4: In a circle with a radius of 10 cm, a tangent is drawn from an external point P. If the tangent segment PT is 6 cm long, find the length of the secant from P to the circle.

Problem 5: In a circle with a radius of 7 cm, a secant is drawn from an external point P such that the external portion of the secant is 15 cm long, and the entire secant is 20 cm long. Calculate the length of the tangent segment from point P to the circle.

Problem 6: In a circle with a radius of 12 cm, point P is located 19 cm away from the center O. A tangent is drawn from point P to the circle, and it touches the circle at point T. Calculate the length of PT.

Tangent Secant Theorem – FAQs

1. What is Tangent?

The line that intersects the circle at only one point is called as the tangent of the circle.

2. Define Secant.

Secant is the line which intersect across the circle at exactly two points.

3. What is the Difference between Tangent and Secant?

The difference between tangent and secant is that tangent is the line which cuts the circle at only one point whereas the secant is the line which cuts across the circle at two distinct points.

4. What is the Statement of the Tangent Secant Theorem?

The tangent secant theorem states that when a tangent and secant are drawn from same external point then, the square of the length of the tangent is equal to the product of the total length of the secant and the length of exterior part of secant.

5. How do we Proof Tangent Secant Theorem?

To prove the tangent secant theorem, we use the properties of the right-angled triangle and pythagoras theorem.

6. What is the Formula for the Tangent Secant Theorem?

The formula for tangent secant theorem is AC × AD = AB².

Where,

• A is the external point to circle,
• B is the point of tangency, and
• C and D are the point of intersection of secant.